Question

How do I find the limit of a convergent sequence?

Answer 1

In general, there is no process that gives you the limit of any convergent sequence. That does not mean, however, that limits cannot be found.

For example, take the sequence `a_n = 1/n`. It's easily seen that `lim_(n to oo) a_n = 0` (just note that, for any positive number `epsilon` close to `0`, we can find a value of `n` big enough such that `1/n < epsilon`).

Generally, the question of the limit of a convergent sequence is not trivial at all, and there are many examples of problems of that kind that went unresolved for years or are still to be resolved.

The question of whether a sequence is convergent or not is easier to answer, even without knowing it's limit (for the case of a convergent sequence), due to Cauchy's criterion.

One famous example of a enduring question is the Basel problem. It consists of the following:

What is the value of the infinite sum of the reciprocals of the squares of the natural numbers?

Or equivalently:

`sum_(k=1)^(oo) 1/(k^2) = lim_(n to oo) sum_(k=1)^(n) 1/(k^2)=?`

We can reformulate this problem in terms of sequences, by defining:

`s_n=sum_(k=1)^(n) 1/(k^2)`

Then the question becomes a problem of finding the limit of a sequence:

`lim_(n to oo) s_n = ?`

It was first posed in 1644 and was solved by Leonhard Euler only in 1735 (91 yeas later), using Taylor polynomials to repesent functions.