Question

How do I find the limit of a series?

Answer 1

There is no general method to do this.

Consider the following definition considering a series of terms `a_k`:

`s_n = sum_(k=0)^n a_k`

This is clearly a sequence, called the sequence of partial sums of the series.

The question of determining the limit of the series

`sum_(k=0)^oo a_k = lim_(n to oo) sum_(k=0)^n a_k = ?`

then becomes a question of determining the limit of a sequence:

`lim_(n to oo) sum_(k=0)^n a_k = lim_(n to oo) s_n = ?`

which does not have a trivial answer.

Some limits can be determined using certain tricks, like the geometric series of initial term `a` and ratio `r` such that `-1 < r < 1`, whose limit can be determined by a simple algebraic trick:

`lim_(n to oo) s_n = s = a + ar + ar^2 + cdots + ar^n + cdots`
`rs = ar + ar^2 + ar^3 + cdots + ar^n + cdots`
`s - rs = a => s = a/(1-r)`

or the the series `sum_(k=0)^oo 1/(k^2)`, whose limit can be found using Fourier Series (this question was called the Basel Problem).