(April 12, 2009 and May 3, 2009)

 For each positive integer n, the following nth partial sums are found in a closed-form.

• .

• .

•  .

• . 

Note that k starts at 2.  As n tends to infinity, it is easy to see this series converges to 1/4,

. 

Also, there exists another series with a different sign of k, namely

.

• . (correct)

• .

• .

• .

• .

• .

• .

The last finite series is a beautiful identity in which both sides are equal for each positive integer n.  It converges to 25/576 as n tends to infinity. In addition, the series can be re-indexed, namely 

or

.

Other finite series:

• Finite Alternative Odd Power Series
• Finite Odd Power Series
• Repeated Power Series
• Sum Of Partial Factorials

Main SMS