How do you evaluate an infinite series?

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Answer 1 · 2018-04-08T14:20:00

See below

There are different types of series, to what use different methods of evaluating

For example a converging geometric series:

$a+ ar + ar^2 + ar^3 + ... + ar^k = sum_(n=1) ^(k) ar^(n-1)$

where $sum_(n=1) ^(k) ar^(n-1) = (a(1-r^k)) / (1-r)$

Assuming $|r| < 1$ we can let $k to oo$ for infinite series to be evaluated ...

$lim_(k to oo ) sum_(n=1) ^(k) ar^(n-1) = lim_(k to oo ) ( a(1-r^k) )/(1-k)$

as $k to oo$ , $r^k to 0$ as $|r|< 1$

$=> sum_(n=1) ^(oo) ar^(n-1) = a/(1-r)$

but there are other series what can be approached with tricks!

Take $1/6 + 1/12 + 1/20 + 1/30 + ...$

After consideration we can recognise this is the same as...

$(1/2 - 1/3 ) + ( 1/3 - 1/4) + (1/4 - 1/5 ) + ...$

$(1/2 cancel(- 1/3) ) + (cancel( 1/3) cancel(- 1/4)) + (cancel(1/4) cancel(- 1/5) ) + ...$

$=1/2$

There are also other infinite series that you can remember, and may be able to prove, a like:

$e^x = 1 + x + x^2 /(2!) + x^3 / (3!) + ... = sum_(n=0) ^oo x^n / (n!)$

There are many others, where there insist one set way of computing infinite series, there are many!