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- Telescoping series - Wikipedia
An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, De dimensione parabolae
- Telescoping Series | Calculus II - Lumen Learning
We notice that the middle terms cancel each other out, leaving only the first and last terms In a sense, the series collapses like a spyglass with tubes that disappear into each other to shorten the telescope For this reason, we call a series that has this property a telescoping series
- Telescoping Series - Sum | Brilliant Math Science Wiki
This is comparable to a collapsible telescope, in which the long spyglass is easily retracted into a small instrument that fits into your pocket As you work through Arron's telescoping series investigation, you would realize that for the series u k = 1 k (k + 1) uk = k(k+1)1 and terms t k = 1 k tk = k1, we have u k = t k t k + 1 uk = tk −tk+
- Telescoping Sum -- from Wolfram MathWorld
A telescoping sum is sum in which subsequent terms cancel each other, leaving only initial and final terms For example, S = sum_ (i=1)^ (n-1) (a_i-a_ (i+1)) (1) = (a_1-a_2)+ (a_2-a_3)+ + (a_ (n-2)-a_ (n-1))+ (a_ (n-1)-a_n) (2) = (a_1-a_n) (3) is a telescoping sum
- Telescoping Series - Matherama
This expression provides the simplified sum of the series by leveraging the telescoping principle, where intermediate terms cancel out, leaving only the initial and final terms of the sequence
- Telescoping series - AoPS Wiki - Art of Problem Solving
In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation This is often done by using a form of for some expression Derive the formula for the sum of the first counting numbers We wish to write for some expression This expression is as
- How to find the sum of a telescoping series - Krista King Math
To determine whether a series is telescoping, we’ll need to calculate at least the first few terms to see whether the middle terms start canceling with each other
- A Useful Mathematical Trick, Telescoping Series, and the In nite Sum of . . .
Brie y, a telescoping series is a sum that is characterized by partial sums (called telescoping sums) that contain pairs of consecutive terms which cancel each other, leaving only the rst and nal terms [8] This cancellation of adjacent terms is whimsically referred to as "collapsing the telescope"
- University of South Carolina
This is an example of a telescoping sum: Because of all the cancellations, the sum collapses (like a pirate's collapsing telescope) into just two terms Figure 3 illustrates Example 6 by show- ing the graphs of the sequence of terms + and the sequence {sn}of partial sums
- Telescoping Series - Oregon State University
So, the sum of the series, which is the limit of the partial sums, is 1 You do have to be careful; not every telescoping series converges Look at the following series: You might at first think that all of the terms will cancel, and you will be left with just 1 as the sum But take a look at the partial sums:
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