# Infinite Series

Arithmetic and Geometric progressions.

### Infinite Series

Prove that:
$$\sum_{x=1}^{\infty }\frac{1}{n^{x}} = \frac{1}{n-1}$$

$$\sum_{x=1}^{\infty }\frac{1}{(-n)^{x}} = -\frac{1}{n+1}$$

with all $$n$$ $$(n \ne 0,n \in ℕ)$$
Guest

### Re: Infinite Series

Guest wrote:Prove that:
$$\sum_{x=1}^{\infty }\frac{1}{n^{x}} = \frac{1}{n-1}$$

That isn't necessarily true! it requires that |n|< 1 and usually (not always) "n" represents a positive integer.
Do you know anything about "geometric series"?

A geometric series is one of the form $$\sum a^n$$. It can be shown that for |a|< 1 that series converges. Suppose it converges to "S". That is, S= 1+ a+ a^2+ .... + a^n+ ... Then S- 1= a+ a^2+ ...+ a^n+ ...= a(1+ a+ ...+ a^(n-1)+ ...). But that last sum, since it goes to infinity, is again S. S- 1= aS so S- aS= (1- a)S= 1 and then S= 1/(1- a).

$$\sum_{x=1}^{\infty }\frac{1}{(-n)^{x}} = -\frac{1}{n+1}$$

with all $$n$$ $$(n \ne 0,n \in ℕ)$$

Let a, above, be -n instead of n.

HallsofIvy

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