Question

How do you evaluate an infinite series?

Answer 1

See below

Explanation:

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!