Section 10.5

Special Series

For most series, we can only tell if they converge. But for a few special types—Geometric and Telescoping—we can find the exact value of the sum.

Geometric Series

A geometric series has a constant ratio between terms.

Form 1 (Starts at

)

Form 2 (Starts at

)

Convergence Test

Convergesif . sum is .
Divergesif .

Example: Messy Geometric Series

Find the sum of .

Rewrite to identify and :

Since , it converges. Apply formula:

Telescoping Series

A series where most of the middle terms in the Partial Sum cancel out, leaving only the first and last parts.

How it Works (Partial Fractions)

Write out the Partial Sum :

Cancel the middle terms:

Take the limit:

The Harmonic Series

The Harmonic Series is a famous divergent series. Even though the terms go to zero, they don't do it "fast enough".

We will prove this in the next section using the Integral Test.

Rules of Convergence

Multiplying by Constants

If diverges, then also diverges (for ).

Stripping Terms

Adding or removing a finite number of terms does not change whether a series converges or diverges.

converges,

also converges.

Worked Examples

Example 1: Finding the Sum of a Geometric Series

Find the value of .

Step 1: Rewrite Powers

Split the exponents to isolate :

Step 2: Identify Form

Simplify to match :

Conclusion

Here, . Since , the series Diverges.

Example 2: Telescoping Series Calculation

Determine the sum of .

Step 1: Write Partial Sums

Step 2: Cancel Terms

Middle terms cancel, leaving the first part of the first term and the last part of the last term: