Infinite Series
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Key Questions

It is very tough to answer such a general question, but I will give it a shot. Infinite series allow us to add up infinitely many terms, so it is suitable for representing something that keeps on going forever; for example, a geometric series can be used to find a fraction equivalent to any given repeating decimal such as:
#3.333...# by splitting into individual decimals,
#=3+0.3+0.03+0.003+cdots# by rewriting into a form of geometric series,
#=3+3(1/10)+3(1/10)^2+3(1/10)^3+cdots# by using the formula for the sum of geometric series,
#=3/{11/10}=10/3# The knowledge of geometric series helped us find the fraction fairly easily. I hope that this was helpful.

Short answer:
The key step is to determine if the term a_n in the infinite series is one side bounded or not (in the limit).Long answer:
Assume that the infinite series is given by#{a_n}_{n=1}^{infty}# , i.e.,#a_n# is the nth term in the series.
1. First step is to check if the term is bounded by a positive finite value for all n
2. Second, check if the term reaches a finite constant in the limit i.e.,
#lim_{n > infty} a_n = c #
More precisely, for a given#epsilon > 0# (a very small positive real number) there exists a integer N such that
#a_n  c < epsilon # if# n > N # .
This basically means#a_n# approaches c as n increases.If a series satisfies this test, then it is convergent.
For example#a_n = (1+2/n)/(2+3/n^2) # is convergent because, for all n, terms a_n are smaller than or equal to#c = 1/2# .
If this test fails, ie., there is no finite term c by which#a_n# is bounded then the series is divergent.
For instance,#a_n = e^n/n^2# is divergent as its values increases without bound.
Finally if the series is said to oscillate if the value fluctuates between two extremes.
eg.#a_n = (1)^n# .
The values are 1 and 1 for odd and even n's respectively.Reference:
Chaptersection 5.1:
A. Khuri, Advanced calculus with applications in statistics, 2nd ed, Wiley, 2003, pp. 132  134. 
I believe that it is the same as an alternating series. If that is the case, then an oscillating series is a series of the form:
#sum_{n=0}^infty (1)^n b_n# , where#b_n ge 0# .For example, the alternating harmonic series
#sum_{n=1}^infty{(1)^n}/n# is a convergent alternating series.

Here is an example of a telescoping series
#sum_{n=1}^infty(1/n1/{n+1})#
#=(1/11/2)+(1/21/3)+(1/31/4)+cdots#
As you can see above, terms are shifted with some overlapping terms, which reminds us of a telescope. In order to find the sum, we will its partial sum#S_n# first.
#S_n=(1/11/2)+(1/21/3)+cdots+(1/n1/{n+1})#
by cancelling the overlapping terms,
#=11/{n+1}#
Hence, the sume of the infinite series can be found by
#sum_{n=1}^infty(1/n1/{n+1})=lim_{n to infty}S_n=lim_{n to infty}(11/{n+1})=1#
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Tests of Convergence / Divergence

1Geometric Series

2Nth Term Test for Divergence of an Infinite Series

3Direct Comparison Test for Convergence of an Infinite Series

4Ratio Test for Convergence of an Infinite Series

5Integral Test for Convergence of an Infinite Series

6Limit Comparison Test for Convergence of an Infinite Series

7Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

8Infinite Sequences

9Root Test for for Convergence of an Infinite Series

10Infinite Series

11Strategies to Test an Infinite Series for Convergence

12Harmonic Series

13Indeterminate Forms and de L'hospital's Rule

14Partial Sums of Infinite Series