# Convergence of an Infinite series and a related Qn

#### bincybn

##### Member
Dear friends,

$$\displaystyle \sum_{x=1}^{\infty}\frac{1}{x}$$ diverges.

But $$\displaystyle \sum_{x=1}^{\infty}\frac{1}{x^{2}}=\frac{\pi^{2}}{6}$$

How can we prove that $$\displaystyle \sum_{x=1}^{\infty}\left(\frac{1}{x^{\left(1+epsilon\right)}}\right)$$ converges to a finite value?

Bincy.

#### Jameson

Staff member
I preface every post like this that I could very well be wrong, but here is my take on it before someone else can confirm/deny my reasoning or provide a different proof.

If epsilon is an integer then I think induction can prove this.

1) Looking at $$\displaystyle \frac{1}{x^n}$$ you know it converges for n=2. I suppose your question is looking at $$\displaystyle \frac{1}{x^{n+1}}$$, so n=1 is true.

2) Using the ratio test, you can show that if n converges that implies that (n+1) converges.

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#### bincybn

##### Member
I agree whatever you said. But there is a small catch.By epsilon, I meant that a very small real no. like 10^-10. For n>=2, we can prove the convergence of the series. But what about 1<n<2 ? If we can prove the convergence for n=1+ (1+ means epsilon greater than 1), any infinite series of this kind converges for n>1.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
I think you can use the integral test for convergence.