How do you write a sum as a telescoping series?

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.

How do you write a partial sum formula?

Thus the sequence of partial sums is defined by sn=n∑k=1(5k+3), for some value of n. Solving the equation 5n+3=273, we determine that 273 is the 54th term of the sequence.

How do you find the nth term of a telescoping series?

How to find formula for nth partial sum of Telescopic Series

1. ∞∑n=1(1n−1n+1)
2. ∞∑n=1((n+1)−nn(n+1))
3. An−Bn+1=(n+1)−nn(n+1)
4. An+A−Bnn(n+1)=(n+1)−nn(n+1)
5. n(A−B)+An(n+1)=(n+1)−nn(n+1)

How do you find a telescoping series?

The series is telescoping if we can cancel all of the terms in the middle (every term but the first and last). Canceling everything but the first half of the first term and the second half of the last term gives an expression for the series of partial sums.

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 .

What is a partial sum of a series?

A partial sum of an infinite series is the sum of a finite number of consecutive terms beginning with the first term. Each of the results shown above is a partial sum of the series which is associated with the sequence .

How are partial sums of a telescoping series written?

By writing the partial sums of a telescoping series in terms of a partial fractions expansion, we see how the inner terms cancel. This cancellation of the inner terms effectively compresses the partial sum like compressing an extended telescope. If the series converges, we are able to find the value of the infinite series.

Why are partial fractions used in telescoping series?

These patterns will more than often cause mass cancellation, making the problem solvable by hand. Some patterns are harder to find than others. Often, partial fractions are used here in a way which shall be demonstrated later. The benefit of such a series is that it allows us to easily add up the terms, because

When is a telescoping series an infinite series?

However, when a telescoping series converges, we can find the numerical value of the infinite series. For a collector of numbers, this is way cool! An infinite series is the sum of an infinite number of terms. For example, 1 + 1/2 + 1/3 + 1/4 + … (out to infinity) is an infinite series.