Show that the series 1 + 2x/5 + 3x^2/25 + 4x^3/125 + ....is convergent if -5<x<5 and for no other value of x.

To determine the convergence of the given series, we can use the ratio test. The ratio test states that if the absolute value of the ratio of consecutive terms of a series approaches a value less than 1 as the number of terms increases, then the series converges.

Let's apply the ratio test to our series:

First, let's find the ratio between consecutive terms. The ratio between the (n+1)th and nth term is given by:

R = [(n+1) * x^n+1 / (n * x^n))] * (25 / 25)

We multiplied by (25 / 25) to simplify the expression and keep the denominator consistent. Now, simplify the expression:

R = [(n+1) * x^n+1 * 25] / [(n * x^n) * 25]
= (n+1) * x^n+1 / (n * x^n)

Next, take the absolute value of the ratio:

|R| = |(n+1) * x^n+1 / (n * x^n)|

For convergence, we need |R| < 1. So we'll set up the inequality:

|(n+1) * x^n+1 / (n * x^n)| < 1

Now, let's analyze this inequality:

1. Start by taking the limit as the number of terms approaches infinity. We consider the limit to check what happens as the terms get larger and larger.

lim(n→∞) |(n+1) * x^n+1 / (n * x^n)| < 1

2. Next, simplify the expression:

lim(n→∞) |(n+1) * x^n+1 / (n * x^n)|
= lim(n→∞) |(n+1) / n| * |x|^(n+1-n)
= |x| * lim(n→∞) (1 + (1/n))

3. Notice that (1 + (1/n)) approaches 1 as n approaches infinity. Therefore, the limit becomes:

lim(n→∞) |x| * (1 + (1/n))
= |x|

4. Since |x| must be less than 1 for the series to converge, we have the condition:

|x| < 1

Thus, the given series converges if and only if -1 < x < 1.

However, in the problem, it is stated that -5 < x < 5. Therefore, for the given series, it converges if -5 < x < 5 and for no other value of x.