Understanding the nth Term Test for Divergence in Series

kamranonline
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I have a problem understanding the divergence of series.. There is a n-th term test that u first apply on the general term of the series and if its limit is not equal 0 then the series is divergent.. When i apply that test sometime i get it wrong and sometime not.. When can i apply this test? is there any special conditions for it? or it should always work.
 
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Please show us examples of what you mean by getting it wrong. The nth term test can tell you if a series diverges, but not all series that diverge fail the nth term test.
 

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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