Part 1: Power and geometric series

A power series is a sum with the general form

1

This is very similar to the general form of a geometric series

2

This is because a geometric series is simply a case of a power series where is a constant and . The expanded form of a geometric series is shown below.

3

The sum of a geometric series can also be written as follows.

4

Proof:

Let

Example 1:

What is the sum of the following?

Using equation 4

Equation 4 can be used to find the sum of a geometric series with terms, or a finite geometric series. How can it be used to find the sum of an infinite geometric series? When dealing with the sum of an infinite series, we must consider whether the series is divergent or convergent.

Part 2: Convergence and Divergence

The sum of an infinite series can be evaluated by observing the trend of the partial sums. If the partial sums tend towards a definite real number, then the series is convergent, and the number is the sum of the series. If the partial sums tend towards or , then the series is divergent, and the sum of the series is or respectively. Finally, is the partial sums do not tend towards any number or towards or , then the series is also divergent and does not have a sum.

Example 2:

The series

Has the following partial sums

Fig 1 ~ Graph of first 15 partial sums

The partial sums tend towards 2, therefore the sum of the infinite series is 2

The series

Has the following partial sums

Fig 2 ~ Graph of first 15 partial sums

The partial sums increase at an increasing rate, therefore the sum the infinite series is .

The series

Has the following partial sums

The partial sums tend towards infinite, but oscillate between positive and negative, therefore this series has no sum.

The examples above show that the sum of an infinite geometric series can only be determined if the series is convergent. For a geometric series to converge, the absolute value of the common ratio, , must be less than one such that

5

6

Example 3:

Find

Using equation 6

Part 3: Ratio test

The ratio test is a general method to determine to a certain extent if an infinite series converges of diverges. The ratio test looks at the limit as of the absolute value of the ratio of two consecutive terms.

7

IF

The series is convergent

8

IF

The series is divergent

9

IF

The ratio test is inconclusive and the nature of the series must be evaluated through other means

10

For a geometric series, using the ratio test, what values of would result in a converging series? For a converging series, has to fit in equation 8

Thus, a geometric series is convergent when

This is also known as the interval of convergence for a geometric series. The interval of convergence is the range of values where the series is convergent. Using the geometric series as an example, the function

Fig 4 ~ Graph of showing the interval of convergence

Exists only within the interval of convergence.

Example 4:

Find the domain of the function

exists when is convergent. Using the ratio test 8

We must also consider the ratio test 10

When

This is the harmonic series and is divergent

When

This is an alternating harmonic series and is convergent.

Therefore, the interval of convergence of is

The domain of is

Part 4: Taylor series

The Taylor series is a power series that can be used to approximate functions. It has the general form

11

12

Where

is the Taylor series expansion of centred at c

Deriving the Taylor series:

Let be an approximation of around a constant

For the first term

For the second term, we want the first derivative of to be the same as the first derivative of at .

This gives the equation

Next, we want the second derivative of to be the same as the second derivative of at .

This gives the equation

This continues on as per equation 11

Example 5:

Given , , , , . Approximate .

We can form a Taylor series approximation for centred around . Using formula 12

The graphs below show the Taylor series approximations for with different numbers of terms.

The Taylor approximation is in orange while the function is in black.

3 terms

4 terms

5 terms

As can be seen from the table above, adding more terms to the series increases the accuracy of the approximation, so in theory we would want to add an infinite number of terms. Of course, a Taylor approximation can only approximate a function about its domain. To find the domain of the Taylor series, we can use the ratio test 8.

Therefore, the domain of any Taylor series is

Part 5: Application of Taylor series approximations

The Taylor series is commonly used to approximate functions and make them simpler to compute. Some common functions to approximate are , , and . These functions are approximated using the Maclaurin series, which is a form of the Taylor series centred around .

13

14

The approximations are as follows

15

16

17

18

19

Approximating these functions is useful for computing as they allow complicated functions, like the sinusoidal functions, to be approximated using simple arithmetic. A calculator today can process about 5 million operations per second. The approximation for contains

20

Calculations where is the number of terms in the approximation. Therefore, a Taylor series approximation of programmed into a calculator could be approximated in under 0.001 seconds if

Fig 6 ~ 42 term Taylor approximation of

However, after a certain value away from 0, the Taylor approximation diverges from . We can calculate how accurate our approximation is, and the interval of which it is accurate, using the remainder function.

The remainder function, ,is defined as the difference between a function, and its approximation, , of degree , centred at .

21

As per the properties of the Taylor series

:

However, when we take the derivative of

22

This is because taking the derivative of a polynomial of degree is . Given that the end goal is to limit the error of our approximation over an interval from to , we could consider limiting the derivative of the remainder function. This can be done by looking at the derivative of , which must be bounded by some value, .

23

From equation 21

Integrating with respect to

Since the absolute value of a function’s integral is less than the integral of the function’s absolute value.

To minimize , using the properties of the Taylor series

Hence

Integrating again

Integrating once more

The pattern continues such that

24

If we take the example of our calculator, we could use equations 23 and 24 to find something such as the interval where the error of approximation is less than .

Using equation 22

Substituting the other values into equation 24 gives

This is a reasonable range, but I think a calculator can do better.

Let’s have the new calculator be able to approximate the value of with an error less than for any . Using equation 24

However, using equation 20, the calculator will have to do about

Calculations and will take about seconds to compute.

Conclusion

In summary, the Taylor series apPart 1: Power and geometric series

A power series is a sum with the general form

1

This is very similar to the general form of a geometric series

2

This is because a geometric series is simply a case of a power series where is a constant and . The expanded form of a geometric series is shown below.

3

The sum of a geometric series can also be written as follows.

4

Proof:

Let

Example 1:

What is the sum of the following?

Using equation 4

Equation 4 can be used to find the sum of a geometric series with terms, or a finite geometric series. How can it be used to find the sum of an infinite geometric series? When dealing with the sum of an infinite series, we must consider whether the series is divergent or convergent.

Part 2: Convergence and Divergence

The sum of an infinite series can be evaluated by observing the trend of the partial sums. If the partial sums tend towards a definite real number, then the series is convergent, and the number is the sum of the series. If the partial sums tend towards or , then the series is divergent, and the sum of the series is or respectively. Finally, is the partial sums do not tend towards any number or towards or , then the series is also divergent and does not have a sum.

Example 2:

The series

Has the following partial sums

Fig 1 ~ Graph of first 15 partial sums

The partial sums tend towards 2, therefore the sum of the infinite series is 2

The series

Has the following partial sums

Fig 2 ~ Graph of first 15 partial sums

The partial sums increase at an increasing rate, therefore the sum the infinite series is .

The series

Has the following partial sums

The partial sums tend towards infinite, but oscillate between positive and negative, therefore this series has no sum.

The examples above show that the sum of an infinite geometric series can only be determined if the series is convergent. For a geometric series to converge, the absolute value of the common ratio, , must be less than one such that

5

6

Example 3:

Find

Using equation 6

Part 3: Ratio test

The ratio test is a general method to determine to a certain extent if an infinite series converges of diverges. The ratio test looks at the limit as of the absolute value of the ratio of two consecutive terms.

7

IF

The series is convergent

8

IF

The series is divergent

9

IF

The ratio test is inconclusive and the nature of the series must be evaluated through other means

10

For a geometric series, using the ratio test, what values of would result in a converging series? For a converging series, has to fit in equation 8

Thus, a geometric series is convergent when

This is also known as the interval of convergence for a geometric series. The interval of convergence is the range of values where the series is convergent. Using the geometric series as an example, the function

Fig 4 ~ Graph of showing the interval of convergence

Exists only within the interval of convergence.

Example 4:

Find the domain of the function

exists when is convergent. Using the ratio test 8

We must also consider the ratio test 10

When

This is the harmonic series and is divergent

When

This is an alternating harmonic series and is convergent.

Therefore, the interval of convergence of is

The domain of is

Part 4: Taylor series

The Taylor series is a power series that can be used to approximate functions. It has the general form

11

12

Where

is the Taylor series expansion of centred at c

Deriving the Taylor series:

Let be an approximation of around a constant

For the first term

For the second term, we want the first derivative of to be the same as the first derivative of at .

This gives the equation

Next, we want the second derivative of to be the same as the second derivative of at .

This gives the equation

This continues on as per equation 11

Example 5:

Given , , , , . Approximate .

We can form a Taylor series approximation for centred around . Using formula 12

The graphs below show the Taylor series approximations for with different numbers of terms.

The Taylor approximation is in orange while the function is in black.

3 terms

4 terms

5 terms

As can be seen from the table above, adding more terms to the series increases the accuracy of the approximation, so in theory we would want to add an infinite number of terms. Of course, a Taylor approximation can only approximate a function about its domain. To find the domain of the Taylor series, we can use the ratio test 8.

Therefore, the domain of any Taylor series is

Part 5: Application of Taylor series approximations

The Taylor series is commonly used to approximate functions and make them simpler to compute. Some common functions to approximate are , , and . These functions are approximated using the Maclaurin series, which is a form of the Taylor series centred around .

13

14

The approximations are as follows

15

16

17

18

19

Approximating these functions is useful for computing as they allow complicated functions, like the sinusoidal functions, to be approximated using simple arithmetic. A calculator today can process about 5 million operations per second. The approximation for contains

20

Calculations where is the number of terms in the approximation. Therefore, a Taylor series approximation of programmed into a calculator could be approximated in under 0.001 seconds if

Fig 6 ~ 42 term Taylor approximation of

However, after a certain value away from 0, the Taylor approximation diverges from . We can calculate how accurate our approximation is, and the interval of which it is accurate, using the remainder function.

The remainder function, ,is defined as the difference between a function, and its approximation, , of degree , centred at .

21

As per the properties of the Taylor series

:

However, when we take the derivative of

22

This is because taking the derivative of a polynomial of degree is . Given that the end goal is to limit the error of our approximation over an interval from to , we could consider limiting the derivative of the remainder function. This can be done by looking at the derivative of , which must be bounded by some value, .

23

From equation 21

Integrating with respect to

Since the absolute value of a function’s integral is less than the integral of the function’s absolute value.

To minimize , using the properties of the Taylor series

Hence

Integrating again

Integrating once more

The pattern continues such that

24

If we take the example of our calculator, we could use equations 23 and 24 to find something such as the interval where the error of approximation is less than .

Using equation 22

Substituting the other values into equation 24 gives

This is a reasonable range, but I think a calculator can do better.

Let’s have the new calculator be able to approximate the value of with an error less than for any . Using equation 24

However, using equation 20, the calculator will have to do about

Calculations and will take about seconds to compute.

Conclusion

In summary, the Taylor series approximation of a function allows the function to be broken up into a series of simpler arithmetic operations that can be programmed into something like a calculator. Increasing the number of terms will increase the accuracy of the approximation but at the cost of efficiency.

To improve the efficiency, programs must look into the individual functions to come up with a solution. In the case of , since it repeats itself at intervals of , an initial operation can be done to find the remainder of . This will require the approximated to only be accurate to the desired degree for any , significantly reducing the number of operations required to approximate the function.proximation of a function allows the function to be broken up into a series of simpler arithmetic operations that can be programmed into something like a calculator. Increasing the number of terms will increase the accuracy of the approximation but at the cost of efficiency.

To improve the efficiency, programs must look into the individual functions to come up with a solution. In the case of , since it repeats itself at intervals of , an initial operation can be done to find the remainder of . This will require the approximated to only be accurate to the desired degree for any , significantly reducing the number of operations required to approximate the function.