Investigating the Convergence of Series: Sn = 5-1/n

KTiaam
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Homework Statement



The nth partial sum of the series
∞
Ʃ an
n=1


is given Sn = 5-1/n


Determine weather the series is convergent or divergent



The Attempt at a Solution



Looked in my book on how to do this one.
couldn't find anything on it.

What i was thinking was find the sum of all n's and finding a pattern and use that as an
however it didnt work, so I am stuck.
 
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KTiaam said:

Homework Statement



The nth partial sum of the series
∞
Ʃ an
n=1


is given Sn = 5-1/n


Determine weather the series is convergent or divergent



The Attempt at a Solution



Looked in my book on how to do this one.
couldn't find anything on it.

What i was thinking was find the sum of all n's and finding a pattern and use that as an
however it didnt work, so I am stuck.

The sum of the series is DEFINED to be the limit of the partial sums. There's not much else to know. What is it?
 

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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