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Differential Calculus

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16 Mar 2008 19:04:35 IST

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Limit of a sequence

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$\text {Define a sequence as follows:} \\ \\ a_n = \frac {1} {10^{n^2}}} \\ \\ \text{Is} \ \lim_{n \to \infty} \sum_{n=1}^\infty \frac {1} {10^{n^2}} \ \text {a rational number?}$

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gokul subramanian

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16 Mar 2008 19:06:07 IST

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bhatt sir isnt it sigma(1 to n) where n tends to infinity

sandeep ramesh

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16 Mar 2008 19:49:40 IST

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i dont get why it shouldnt be rational maybe

Bcos the summation is done over rational terms only right? Then why shouldnt the sum be rational only?

amaron

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16 Mar 2008 20:29:39 IST

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What is given is a non recurring,non terminating decimal .
Hence not a rational number.

Note that sum of infinite rational terms need not be rational.

e^x = 1 +x +x² /2! ....

is obviously not rational even for rational values of x.

That solves the problem!

sandeep ramesh

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16 Mar 2008 20:38:54 IST

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Is that so mr bhatt

but i think its only lim n----> infinity n not infinity itself so its still a finite number of terms making it rational maybe

Hari Shankar

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16 Mar 2008 21:13:49 IST

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nice work amaron.

I mixed it up a little, it is of course a sum to infinite terms.

If you write it down it looks like 0.100100001..., this is a non-terminating, non-recurring decimal representation, which is typical of an irrational number

sandeep ramesh

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16 Mar 2008 21:49:34 IST

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oh yes nice :)

Hari Shankar

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16 Mar 2008 22:05:01 IST

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I simplified the question a bit. The original one had a +ve integer k instead of 10.

Then too the result hold, for in base k, it will read 0.100100001...

I thought it was a lovely concept to bring here

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