# What is the radius of convergence of #sum_1^oo x/n#?

The series:

is convergent for

Applying the test of ratios:

The series turns into:

As:

and

The Leibniz theorem indicates that the series is convergent.

The series turns into:

Take a look at the partial sum sequence:

and:

that is:

which demonstrates that the series is not convergent since Cauchy's condition is not met.

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The radius of convergence of the series (\sum_{n=1}^{\infty} \frac{x}{n}) can be found using the ratio test.

The ratio test states that for a series (\sum_{n=1}^{\infty} a_n), if (L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|), then:

- If (L < 1), the series converges absolutely.
- If (L > 1) or (L = \infty), the series diverges.
- If (L = 1), the test is inconclusive.

For the series (\sum_{n=1}^{\infty} \frac{x}{n}), (a_n = \frac{x}{n}). Applying the ratio test:

[ L = \lim_{n \to \infty} \left| \frac{\frac{x}{n+1}}{\frac{x}{n}} \right| = \lim_{n \to \infty} \left| \frac{x}{n+1} \cdot \frac{n}{x} \right| = \lim_{n \to \infty} \left| \frac{n}{n+1} \right| = 1 ]

Since (L = 1), the ratio test is inconclusive.

To find the radius of convergence, we can consider the ratio test when (L = 1), which is the boundary case. In this case, the series converges if (\left| \frac{x}{n+1} \cdot \frac{n}{x} \right| < 1). Solving this inequality for (x), we get:

[ \left| \frac{x}{n+1} \cdot \frac{n}{x} \right| < 1 \implies \left| \frac{n}{n+1} \right| < 1 \implies \frac{n}{n+1} < 1 \implies n < n+1 ]

Since (n) can be any positive integer, there is no restriction on (x), meaning the series converges for all real numbers (x).

Therefore, the radius of convergence of the series (\sum_{n=1}^{\infty} \frac{x}{n}) is (\infty).

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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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