# How do you find the interval of convergence #Sigma (-1)^nx^n/n# from #n=[1,oo)#?

The series:

is convergent for

Given the series:

we can use the ratio test by evaluating:

So that the ratio limit is:

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The interval of convergence for the series ( \sum_{n=1}^{\infty} (-1)^n\frac{x^n}{n} ) can be found using the ratio test. The ratio test states that the series converges if the limit as ( n ) approaches infinity of the absolute value of ( \frac{a_{n+1}}{a_n} ) is less than 1. In this case, ( a_n = (-1)^n\frac{x^n}{n} ). Applying the ratio test, you will find that the interval of convergence is ( -1 < x \leq 1 ), including ( x = -1 ) and excluding ( x = 1 ).

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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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