# Thread: Problem of sequence

1. ## Problem of sequence

First of all, please my poor English (I'm currently a French student trying to master mathematical redaction in another language... And I do find it difficult.)
So, I have an exercise I don't know how to handle...

Let (Wn) be a sequence such as W_n= sum from (k=1 to n) of (1/(n+k)^1/2). Prove that for any integer n, (n/2)^1/2<= (Wn)
I tried to prove it using induction:
Let for any positive integer be Pn:" (n/2)^1/2<=(Wn)
Basis: for n= 1, we have W_0= 1/(1+1)^1/2=1/(2)^1/2 , thus (P_0) is confirmed
Then I assumed Pn, and tried to study W(n+1)
Which I'm currently failing at...

Any tips, any idea about how I could resolve this?

Thanks for reading me! (And sorry for not using LaTEX, I'm currently trying to get accustomed to it... :/ )

2. ## Re: Problem of sequence

Each of n terms in $\displaystyle W_n$ is at least $\displaystyle \frac{1}{\sqrt{2n}}$ ...

3. ## Re: Problem of sequence

Hum, I assume you're right (yet I haven't worked out WHY each of the terms of Wn is greater than 1/(V(2n))... Since Pn only assume that Wn is greater than V(n/2). But I'll find the reason why anyway, thank you very much! )

4. ## Re: Problem of sequence

It's easier to give a direct, not an inductive, proof. $\displaystyle \frac{1}{\sqrt{n+k}}\ge\frac{1}{\sqrt{2n}}$ because $\displaystyle n+k\le2n$.

5. ## Re: Problem of sequence

Oh yep, got it! Thank you, the rest followed from it quite easily!