In details

P-series, alternating series and power series


It is a series as follows:

CONVERTS if p> 1

If p = 1, the series

it's called harmonic series and according to the theorem it is divergent.

Alternating series

It looks like this:

Power Series

X power series


(X-c) Power Series

For convenience, let's assume that even when x = 0.

When replacing x by a real number, we get a series of constant terms that can converge or diverge.

In any power series of x, the series always converges to x = 0, because if we replace x per 0 the series comes down to0.

In the power series of (x-c), the series converges to x = c.

To determine the other values ​​of x for which the series converges, the test of reason. Next: Leibniz Test