How do you write a sum as a telescoping series?

A telescoping series is a series where each term u k u_k uk can be written as u k = t k − t k + 1 u_k = t_{k} – t_{k+1} uk=tk−tk+1 for some series t k t_{k} tk.

What is the meaning of telescoping sum?

A sum in which subsequent terms cancel each other, leaving only initial and final terms.

What is telescoping in math?

In mathematics, a telescoping series is a series whose general term can be written as , i.e. the difference of two consecutive terms of a sequence .

How do you find the formula for the nth partial sum of a series?

The nth partial sum of a geometric sequence can be calculated using the first term a1 and common ratio r as follows: Sn=a1(1−rn)1−r. The infinite sum of a geometric sequence can be calculated if the common ratio is a fraction between −1 and 1 (that is |r|<1) as follows: S∞=a11−r. If |r|≥1, then no sum exists.

Why is it called telescoping series?

A telescoping series is any series with terms that cancel out with other terms. It’s called “telescopic” because part of each term is canceled out by a later term, collapsing the series like a folding telescope.

What is the formula for the sum of a geometric series?

To find the sum of a finite geometric series, use the formula, Sn=a1(1−rn)1−r,r≠1 , where n is the number of terms, a1 is the first term and r is the common ratio .

How to find the sum of a telescoping series?

Telescoping series is a series where all terms cancel out except for the first and last one. This makes such series easy to analyze. In this video, we use partial fraction decomposition to find sum of telescoping series. Created by Sal Khan. This is the currently selected item.

How is the telescoping series similar to a telescope?

. This is comparable to a collapsible telescope, in which the long spyglass is easily retracted into a small instrument that fits into your pocket. 1 k − 1 k + 1 = k + 1 k ( k + 1) − k k ( k + 1) = 1 k ( k + 1).

Is the Khan Academy series a telescoping series?

Sal mentioned at the very end of the video that the series was a telescoping series. Now my question with these types of series is that if you had a telescoping series like the one in the video where it’s the summation from n = 1, to n = infinity, wouldn’t the answer always just be the first term of the partial fraction decomposition?

What does the Sigma stand for in telescoping?

Direct link to ArDeeJ’s post “It’s a capital sigma, the greek letter Σ. It stand…” It’s a capital sigma, the greek letter Σ. It stands for summation. The “n = 2” below it tells that the variable n starts with the value 2, and the ∞ above tells that it runs to infinity.