Telescoping Sums in Mathematics

First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as thus to discern the particular laws governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general laws that complete theories by bringing a multitude of apparently very diverse phenomena together under a single governing principle.

Augustin Louis Cauchy

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Telescoping sums are finite sums in which pairs of consecutive terms cancel each other, leaving only few terms (in most of the cases the initial and final terms). This is a challenging portion of algebra that requires the solver to look for patterns. These patterns will more than often cause mass cancellation, making the problem solvable easily. Often, partial fractions are used to solve the problems. Let tn be a sequence of numbers. Then

Mathematically speaking, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. Consider the following example, which is easy and good to start.

Solution. Since it is evident that this is an example of telescoping series, we need to look for patterns. The terms in the denominators of the series are 2, 6, 12, 20, . . . and we can write them as 1 × 2, 2 × 3, 3 × 4, 4 × 5, . . . , 200 × 201. In view of these, we can rewrite the series as

As mentioned earlier, it’s time for partial fractions. The difference between the terms in the denominators is 1 and 1 is there in the numerator. Therefore, a general term looks like

In view of this result, we can easily write

Finally, it follows that

We now consider the following example, slightly different from the previous one. Analyse the terms given in the series and then proceed.

Solution. Since it is evident that this is an example of telescoping series, we need to look for patterns. The terms in the denominators of the series are 4, 28, 70, 130, . . . and we can write them as 1 × 4, 4 × 7, 7 × 10, 10 × 13, . . . , 97 × 100. Satisfy yourself that, this is the correct pattern because 4 is there in the first and second term; 7 is there in the second and third term; and so on. In view of these, we can rewrite the series as

Now, the difference between the terms in the denominators is 3. Therefore, a general term looks like

In view of this result, we can easily write

Finally, it follows that

Solution. Recall the formula (√a + √b) (√a – √b) = ab. Moreover, the numbers inside the square root in each term differ by 1. Again, we need to transform the given sum into a telescoping type sum. To do this, we need to multiply both numerator and denominator by the rationalising factor. Combining all these facts, a general term looks like

In view of this result, we can easily write

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3 comments

  1. Really interesting and insightful. Though we use these often in high school math, I wasn’t aware of the term telescopic sums. Thank you.

    Liked by 1 person

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