Show that $\sum\limits_{n=1}^{32}\frac1{n^2}=1 + \frac{1}{4} +\frac{1}{9} +\dots+ \frac{1}{1024}<2$

Question

Show that $$1 + \frac{1}{4} +\frac{1}{9} +\dots+ \frac{1}{1024} <2$$

I know that the denominators are perfect squares starting from that of $1$ till $32$. Also I know about this identity $$\frac{1}{n(n+1)} > \frac{1}{(n+1)^2} > \frac{1}{(n+1)(n+2)}.$$ But I am not able to implement it Please help me.

score 2 · Answer 1 · 2017-01-10T15:16:52.530

2

We know $$\frac{1}{n(n+1)} > \frac{1}{(n+1)^2} \Rightarrow \frac{1}{n} - \frac{1}{n+1} > \frac{1}{(n+1)^2}$$ Adding for $n =1,2,\cdots,31$, we get, $$\frac{1}{1} - \frac{1}{2} > \frac{1}{4}$$ $$\frac{1}{2}-\frac{1}{3} > \frac{1}{9}$$ $$\vdots$$ $$\frac{1}{31}-\frac{1}{32} > \frac{1}{1024}$$Adding gives us $$\frac{1}{4} +\frac{1}{9} +\cdots +\frac{1}{1024}< 1-\frac{1}{32}$$ $$\Rightarrow 1+\frac{1}{4}+ \cdots+\frac{1}{1024}< 2-\frac{1}{32} < 2$$ Hope it helps.

edited Jan 10 '17 at 15:16

answered Jan 10 '17 at 15:09

Thanks a lot! Well what should we do in case of when series is less than 1.8 I don't think this summing or identity will help – Yami Kanashi Jan 10 '17 at 15:11
Can this help? n/n+1<n+1/n+2 – Yami Kanashi Jan 10 '17 at 15:12
Yeah instead of 2 – Yami Kanashi Jan 10 '17 at 15:13
@LokeshSangewar IF other question THEN new post, please. – Did Jan 10 '17 at 15:13
Well I just need a hint for that not complete answer.. That's why not print a new one.... Well I'll try first – Yami Kanashi Jan 10 '17 at 15:14
@juniven I did not add 1 in the second last expression. – Jan 10 '17 at 15:20
@Rohan Its okay now. You don't need to simplify it since you need to add by 1 both sides – Juniven Acapulco Jan 10 '17 at 15:22
@LokeshSangewar For the smaller bound, take a look here. – Daniel Fischer Jan 10 '17 at 16:03

score 2 · Accepted Answer · answered Jan 10 '17 at 15:33

2

Another way:

$$\begin{align} \sum_{k=1}^{2^5} \frac{1}{k^2} &\leq 1 + \sum_{k=2}^{2^5}\int_{k-1}^{k}\frac{1}{t^2} dt\\ &=1+\int_1^{2^5}\frac{1}{t^2} dt \\ &=2-\frac{1}{2^5}\\ &<2 \end{align}$$

answered Jan 10 '17 at 15:33

Gabriel Romon

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hamam_Abdallah · Answer 3 · 2017-01-10T15:22:00.837

1

For $n>0$, we have $$\frac{1}{(n+1)^2}\leq \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$$

and by sum, $$1+\sum_{n=1}^{31}\frac{1}{(n+1)^2}\leq 2-\frac{1}{32}<2$$

edited Jan 10 '17 at 15:22

answered Jan 10 '17 at 15:16

hamam_Abdallah

62,951

Well since it involves integration... It's kinda complex for me to understand – Yami Kanashi Jan 10 '17 at 15:19
@LokeshSangewar Is it easier now. – hamam_Abdallah Jan 10 '17 at 15:23
Yes.... It is. Lol – Yami Kanashi Jan 10 '17 at 15:25

Show that $\sum\limits_{n=1}^{32}\frac1{n^2}=1 + \frac{1}{4} +\frac{1}{9} +\dots+ \frac{1}{1024}<2$

3 Answers3