W11 - HW Solutions

01 - Modifying a known power series

Consider the power series for .

(a) By modifying the series , write as a power series centered at and determine its radius of convergence.

(b) By modifying the series , write as a power series centered at and determine its radius of convergence.

Solution

(a)

Rewrite in the form .

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Expand out power series.

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Determine radius of convergence.

(b)

Rewrite in the form .

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Expand out power series.

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Determine radius of convergence.

03 - Approximating

Using the series representation of , show that:

Now use the alternating series error bound to approximate to within .

Solution

Write the down series representation of .

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Substitute for to find the series representation for .

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Plug in to find the series for .

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Use the alternating series error bound.

Therefore is an estimate accurate with .

04 - Power series of a derivative

Suppose that a function has power series given by:

The radius of convergence of this series is . What is the power series of and what is its radius of convergence?

Solution

Take the derivative of the sum.

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Determine radius of convergence.

Since the radius of convergence of is 1, we conclude the radius of convergence of .

05 - Finding a power series

Find a power series representation of these functions:

(a)

(b)

Solution

Rewrite in format .

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Plug into geometric series.

Geometric series in .

Plug in .

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Multiply by

(b)

Find the power series of (see example 10-05).

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Multiply by to find the power series of .

06 - Modifying and integrating a power series

(a) Modify the power series for to obtain the power series for .

(b) Now integrate this series to find the power series for .

Solution

(a)

Rewrite as and find the power series.

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(b)

Integrating term by term, we get

11 - Summing a Maclaurin series by guessing its function

For each of these series, identify the function of which it is the Maclaurin series:

(a)

(b)

Now find the total sums of these series:

(c)

(d)

Solution

(a)

Write in the form of known Maclaurin series.

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(b)

Write in the form of known Maclaurin series.

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(c)

Write in the form of known Maclaurin series, where .

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(d)

Write as a product of two Maclaurin series.

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Evaluate series.

14 - Large derivative using pattern of Maclaurin series

Consider the function . Find the value of .

(Hint: find the rule for coefficients of the Maclaurin series of and then plug in 0.)

Solution

Find series for using our known series and plugging in .

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Multiply series by to obtain the series for .

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Note that MacLaurin series of have the form , where . So, .

Compute .

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Compute .

15 - Estimates with alternating series error bound

Without a calculator, estimate (the angle in radians) with an error below .

(Use the error bound formula for alternating series.)

Solution

Write the Maclaurin series of .

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Implement error bound to set up question for .

Find such that , and therefore

Plug in .

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Find when .

The first time is below is when .

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Derive conclusions.

The sum of prior terms equals .

For , there is no term, so , so .

16 - Estimates with alternating series error bound

Find an infinite series representation of and then use your series to estimate this integral to within an error of .

(Use the error bound formula for alternating series.)

Solution

Write the series of the integrand. Note that .

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Compute integral of series.

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Compute terms until coefficient is below .

The first coefficient that is below is . So, the sum of the terms before this will give you an appropriate estimate.

Final answer is .