Chapter 8: Q 55. (page 670)

Explain why the series is not a power series in $x - x_{0}$ .Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge $\sum_{k = 0}^{\infty} \frac{{(\sin x)}^{k}}{k!}$

Short Answer

Expert verified

The value of $x$ for which the series $\sum_{k = 0}^{\infty} \frac{{(\sin x)}^{k}}{k!}$ converges when $x \in R$ .

Step by step solution

Step 1. Given information.

The given power series is $\sum_{k = 0}^{\infty} \frac{{(\sin x)}^{k}}{k!}$

Step 2. Find the values of x for which the given series converge.

Since, the series consists of power of $S i n x$ . So the series in not power series in $x - x_{0}$ .

Now,

$b_{k} = \frac{{(\sin x)}^{k}}{k!}$ and $b_{k + 1} = \frac{{(\sin x)}^{k + 1}}{(k + 1)!}$

Thus,

$\lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| = \underset{k \to \infty}{l i m} |\frac{\frac{{(\sin x)}^{k + 1}}{(k + 1)!}}{\frac{{(\sin x)}^{k}}{k!}}| = \underset{k \to \infty}{l i m} |\frac{\sin x}{(k + 1)}|$

So, for $k \to \infty$ the value of limit will always be zero.

Therefore, the value of $x$ for which the series $\sum_{k = 0}^{\infty} \frac{{(\sin x)}^{k}}{k!}$ converges when $x \in R$ .

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Explain why the series is not a power series in $x - x_{0}$ .Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge $\sum_{k = 0}^{\infty} \frac{{(\sin x)}^{k}}{k!}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the values of x for which the given series converge.

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Explain why the series is not a power series in x-x0.Then use the ratio test for absolute convergence to find the values of xfor which the given series converge ∑k=0∞sinxkk!

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the values of x for which the given series converge.

One App. One Place for Learning.

Most popular questions from this chapter

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Explain why the series is not a power series in $x - x_{0}$ .Then use the ratio test for absolute convergence to find the values of $x$ for which the given series converge $\sum_{k = 0}^{\infty} \frac{{(\sin x)}^{k}}{k!}$