harmonic series,.... please help!!

[VINEET]

what is the sum of the following series
1+1/1+1/2+1/3+1/4.......

1+1/1^2+1/2^2+1/3^2+1/4^2......
 
please tell the methgod to solve it...

[VINEET]

#method

[SINGHAM]

1+1/1+1/2+1/3+1/4.......
diverges, by Cauchy condensation.

1+1/1^2+1/2^2+1/3^2+1/4^2......  converges.


Hmmm! METHOD? I don't know which exam asks that? But You can use comparison test or ratio test to find out whether it is convergent or divergent. 1+1/1^2+1/2^2+1/3^2+1/4^2+.....=1+π^2/6 (1+ (pi^2)/6), but don't know the method for it.

[ashutosh]

@singham thanks for that.

But is there a way to find the exact sum for these.

@vineet... I had the same doubt :)

[VINEET]

searched on wikipedia, it says u can test convergence via integral test.

[SINGHAM]

O yes, there are plenty of tests, that says you whether it Converges or Diverges, but if it converges, then how will you find the exact value????

By the way which exam question is this, Vineet?

[VINEET]

its from jnu sss 2010 and 2012 resp.  thanks brother for it help

[VINEET]

# your

[SINGHAM]

Hope this will help you   http://math.cmu.edu/~bwsulliv/basel-problem.pdf

(THOUGH I COULDN"T UNDERSTAND THAT)