1. Show that the following series do not converge:(a)∞Xn=1r n2(n + 1),

To show that the series ∑(n=1 to ∞) r^n / (n^2(n + 1)) does not converge, we can use the divergence test.

The divergence test states that if the limit of the terms of a series does not approach zero as n approaches infinity, then the series diverges.

Let's find the limit of the terms of the series as n approaches infinity.

lim(n→∞) r^n / (n^2(n + 1))

We can rewrite the expression as:

lim(n→∞) (r^n / n^2) / (n + 1)

Now, let's consider the numerator:

lim(n→∞) r^n / n^2

If r > 1, then as n approaches infinity, r^n will also approach infinity. Therefore, the numerator will approach infinity.

If r = 1, then the numerator will approach 1.

If r < 1, then as n approaches infinity, r^n will approach 0. Therefore, the numerator will approach 0.

Now, let's consider the denominator:

lim(n→∞) n + 1

As n approaches infinity, the denominator will approach infinity.

Now, let's put it all together:

lim(n→∞) (r^n / n^2) / (n + 1)

If r > 1, the numerator approaches infinity and the denominator approaches infinity. Therefore, the limit is indeterminate.

If r = 1, the numerator approaches 1 and the denominator approaches infinity. Therefore, the limit is 0.

If r < 1, the numerator approaches 0 and the denominator approaches infinity. Therefore, the limit is 0.

Since the limit of the terms of the series does not approach zero as n approaches infinity for r > 1, the series does not converge.