Proving Cesaro (C,1) Summability of Series

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Homework Statement



Can anyone show that any of these two series is Cesaro (C,1) summable)

a) 1+0-1+1 +0-1+···

and

b) 1-1+2-2+3-3+4-4+···


thanks
 
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Hi errordude, what are you thoughts/attempt?

start with the definition of cesaro summable
 
lanedance said:
Hi errordude, what are you thoughts/attempt?

start with the definition of cesaro summable


yeah i know the def of Cesaro sums.
 

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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