Understanding Series Convergence and Changing n=1 to n=1+k

ptolema
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this is just a general question: do
conv.jpg
both converge to the same limit? is there a general rule for changing n=1 to n=1+k?
 
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Convergence of a series is not affected if you add/remove finitely many terms to it. A more rigorous statement can be found in any of the many elementary books on analysis, for e.g. W. Rudin's Principles of Mathematical Analysis.
 
alright, thanks! i had an intuitive idea, but i wasn't sure if there was something i had to do to an first. makes a lot more sense now
 
Yes. The other way would be to replace n by m, where m = n+k (so that the range of summation changes from 0 to Inf -> k + Inf); Since m is a dummy variable, we can as well call it n.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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