Home > Precalculus > Concepts and Informal Definition of a Limit

How do I find the limit of a series?

Answer 1

Lily Anderson

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).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.