- Topics At a Glance
- Series: This is the Sum That Doesn't End
- Sigma Notation
- Alternating Series
**Convergence of Series**- Finally, Meaning...and Food
**Relationship Between Sequences**- Math-e-magics?
- When Limits of Summation Don't Matter
- Properties of Series
- Special Cases
- Arithmetic Series
- Geometric Series
- Finite Geometric Series
- Infinite Geometric Series
- Decimal Expansion
- Word Problems
- Visualization of Series
- When Limits of Summation Don't Matter
- Tests for Convergence
- The Divergence Test
- The Alternating Series Test
- The Ratio Test
- The Integral Test
- The Comparison Test
- Absolute Convergence vs. Conditional Convergence
- Summary of Tests
- Taylor and Maclaurin Series
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

It turns out that the sequence of terms and the sequence of partial sums of a series are not distant second cousins. They are both born from the same mother series, so they are siblings. Sometimes they look just alike, similar to identical twins. Sometimes they look as different as peas and carrots. They go together, nonetheless.

Look at any series

we can make a sequence out of the terms of the series:

*a*_{1}, *a*_{2}, *a*_{3}, ...

We can also make a sequence out of the partial sums of the series:

*S*_{1}, *S*_{2}, *S*_{3}, ....

These two sequences are different. They won't necessarily converge to the same value. They might not both converge. However, they are related in one key way.

*If the sequence of partial sums converges, then the sequence of terms must converge to zero.*

We know that saying, "the sequence of partial sums converges," is the same as saying, "the series converges." Here's a another mind-blowing fact.

*If a series converges, then its terms must converge to zero.*

Mathematician and magician are two very different professions. Because we don't like waving our hands, turning around three times and spitting to prove things, we are going to show both of these are true using math.