## tanjung Group Title simplify ... one year ago one year ago

1. tanjung

|dw:1356426308338:dw|

2. UnkleRhaukus

$=\sum\limits_{n=4}^{25}\frac{1}{n^2+1}$

3. tanjung

i think, the problem wants a rational number

4. UnkleRhaukus

hmm http://www.wolframalpha.com/input/?i=sum+%281%2F%7Bn%5E2%2B1%7D%29+from+n%3D4+to+25 well it is rational, , ,

5. tanjung

if without wolfram ?

6. UnkleRhaukus

if we have interpreted the question correctly, and the wolfram result is the one you are aiming towards, then this wont be a nice problem ,

7. tanjung

hmmm... is there a method to calculating series of : 1/(n^2+1) + 1/((n+1)^2+1) + ... + 1/((n+k)^2+1) , @UnkleRhaukus ?

8. UnkleRhaukus

im not sure ,

9. UnkleRhaukus

what is the context that the problem came from ?

10. tanjung

this is a question from one of group math also, in my fb. the question ask to calculate or simplify it

11. wio

The real question is if there is a simplification of: $\sum \frac{1}{n^2+1}$

12. wio

Like, maybe partial fraction decomposition could help us?

13. wio

I'm not familiar with this series actually. I'm not sure how to find it's partial sum.

14. mukushla

@calculusfunctions that power series u use is for when $$|x|<1$$

15. Stiwan

@calculusfunctions @mukushla On the topic of the power series, there is this equation: $\sum_{k=0}^{n}x^k = \frac{1-x^{n+1}}{1-x}$ If we suppose |x| < 1 and let n go to infinity, x^(n+1) converges to 0, so the power equation equation becomes $\sum_{k=0}^{\infty} x^k = \frac{1}{1-x}$ So you can use the power series only to calculate infinite series if |x|<1, but as the sum is finite in this example we could use it as well, if we would be able find a constant factor x so that $\frac{1}{n²+1} * x =\frac{1}{(n+1)²+1} \forall n \in [4,24] \subseteq \mathbb{N}$ You see immediately that this factor x exists but it is not CONSTANT (it depends on n), so we can't use the power series here. @tanjung I'm pretty sure that the answer UncleRauhkus gave you is the correct one, because I can't see the point of an exercise that would require you to do any calculations beside that, because it certainly would not "simplify" the matter.

16. mukushla

i meant that power series will not be useful for this problem and there is no way to simplify that expression...if im not wrong

17. calculusfunctions

Yes @mukushla and @sitwan you're both right. I know that, that's why I didn't proceed further. I was just thinking out loud.