Math
Alternating Series Test
Archimedean Property
Bolzano Weismann Theorem
Cauchy Sequences
Fundamental Trigonometric Limit
Limits at Infinity and Asymtotes
Limits of a Function
Limits of a Sequence
Limits of Rational Functions
LUB GLB
Monotone Convergence Theorem (MCT)
Monotonic Sequences
Peak Points
Rational Function
Series
Squeeze Theorem
Subsequence
Tail

Resources https://courses.lumenlearning.com/calculus2/chapter/telescoping-series/

Given a sequence , the formal sum

is called a series. (The series is called formal because we have not yet given it a meaning numerically.)

The 's are called the terms of the series. For each term , is called the index of the term.
We will denote the series by

Sk = \sum{n=1}^{k} a_n

\sum{n=1}^{\infty} a_n = L
$$and assign the sum this value. Otherwise, we say that the series $\sum{n=1}^{\infty} a_n$ diverges.

Telescoping Series
...

A telescoping series is a series in which most of the terms cancel in each of the partial sums, leaving only some of the first terms and some of the last terms.

Example:

Another Trigonometric Example:

Geometric Series
...

A geometric series is a series of the form

Assuming (is trivial)
Given that

Multiplying this with we get
Now, let's subtract the second equation from the first:
Which simplifies to:
And as long as : If then goes to 0 as . Therefore in this case converges

If then diverges

Divergence Test
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The Divergence Test states that if a series converges, then

Note this is a statement and not statement

The Divergence test is it’s contrapositive to check if series diverges

Taking the Contrapositive

Proof
If converges then .

Harmonic series proves that this isn’t true Proofs

Interactive Graph
Table Of Contents

Telescoping Series

Geometric Series

Divergence Test