# The series #sum_(n=1)^oo x^n/10^n # converges for #|x| lt beta#, find #beta#?

We can apply d'Alembert's ratio test:

Suppose that;

Then

if L < 1 then the series converges absolutely; if L > 1 then the series is divergent; if L = 1 or the limit fails to exist the test is inconclusive.

Our series is;

So our test limit is:

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The series (\sum_{n=1}^{\infty} \frac{x^n}{10^n}) converges if (|x| < 10). Therefore, (\beta = 10).

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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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