# What is the interval of convergence of #sum_1^oo sin((pi*n)/2)/n^x #?

The series:

is convergent for

To begin, we observe that:

So:

Since this is an alternating series, convergence occurs when:

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The interval of convergence for the series ( \sum_{n=1}^\infty \frac{\sin\left(\frac{\pi n}{2}\right)}{n^x} ) is ( x > 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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