Chapter 1: Q17P (page 41)

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.
$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{π}{2})}^{2 n}$

Short Answer

Expert verified

The sum of the series, i.e. $\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{π}{2})}^{2 n} = 0$ ,

Step by step solution

Given Information

The given series, i.e. $\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{π}{2})}^{2 n}$ ,

Definition of Sum of a Series

The total of all the terms in a sequence can be written as the sum of a series.

Find the sum of the series

Use the Maclaurin series of cos x,

$\cos (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(x)}^{2 n}$

Further, put $x = \frac{π}{2}$ and simplify as:

$\cos (\frac{\begin{matrix} π \end{matrix}}{2}) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{\begin{matrix} π \end{matrix}}{2})}^{2 n} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{\begin{matrix} π \end{matrix}}{2})}^{2 n} = \cos (\frac{\begin{matrix} π \end{matrix}}{2}) = 0$

Hence, the sum of the series, i.e. $\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{\begin{matrix} π \end{matrix}}{2})}^{2 n} = 0$ ,

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Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.
$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{π}{2})}^{2 n}$

Short Answer

Step by step solution

Given Information

Definition of Sum of a Series

Find the sum of the series

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Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.∑n=0∞(-1)n(2n)!(π2)2n

Short Answer

Step by step solution

Given Information

Definition of Sum of a Series

Find the sum of the series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.
$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} {(\frac{π}{2})}^{2 n}$