Telescoping Series Calculator
Calculate telescoping series sums where consecutive terms cancel out.
Series Type
Series:
∑ [1/n - 1/(n+1)] from n=1 to n=10
Telescoping Sum
0.90909091
Simplification
= 1/1 - 1/11
Term Cancellation
Remaining: 1/1, -1/11
Cancelled: 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10
Individual Terms (first 10)
n=1:0.500000
n=2:0.166667
n=3:0.083333
n=4:0.050000
n=5:0.033333
n=6:0.023810
n=7:0.017857
n=8:0.013889
n=9:0.011111
n=10:0.009091
What is a Telescoping Series?
A telescoping series is a series where most terms cancel with preceding or following terms, leaving only a few terms to evaluate. This makes finding the sum much simpler.
Example: ∑(1/n - 1/(n+1))
(1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ...
= 1/1 - 1/(n+1) = 1 - 1/(n+1)
Key Technique
Use partial fractions to decompose terms into differences that telescope.