Telescoping Series

A telescoping series is a series where each term $u_k$ can be written as $u_k = t_{k} - t_{k+1}$ for some series $t_{k}$ . The benefit of such a series, is that it allows us to easily add up the terms, because $u_1 + u_2 + u_3 + \ldots + u_n = (t_1 - t_2) + (t_2 - t_3) + (t_3 - t_4 ) + \ldots + (t_n - t_{n+1} ) = t_1 - t_ {n+1}.$

Observe that most of the $t_k$ terms cancel out with their counterparts in other brackets, and hence we are only left with $t_1 - t_{n+1}$ . 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 $u_k = \frac{1}{k(k+1)}$ and terms $t_k = \frac{ 1}{k}$ , we have $u_k = t_k - t_{k+1}$ since

$\frac{ 1} { k} - \frac{ 1} { k+1} = \frac{ k+1} { k(k+1) } - \frac{ k} { k(k+1) } = \frac{ 1}{ k (k+1) } .$

As such, we have our telescoping series. This allows us to conclude that

$\sum_{i=1}^n \frac{1}{i(i+1) } = \frac{1}{1} - \frac{ 1}{n+1}.$

In particular, since $\frac{1}{n+1}$ approaches 0 as $n$ gets large, we get that

$\sum_{i=1}^\infty \frac{1}{i(i+1) } = 1.$

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Comments

You may want to update the link to Arron's Set. Here it is: https://brilliant.org/profile/arron-udsft3/sets/telescoping-series-investigation/

Also, typo: "we have $u_k=t_k-t_{k+1}$ "

Daniel Liu - 7 years, 2 months ago

Thanks! Edited :)

Calvin Lin Staff - 7 years, 2 months ago

This method of reduction is known as Partial Fraction. Often useful in integration also.

Niranjan Khanderia - 7 years ago

This is not the method of partial fractions. Please read again to clear doubts.

A Brilliant Member - 6 years, 10 months ago

Partial-Fraction Decomposition is the full name.

Niranjan Khanderia - 6 years, 10 months ago

$(x-4),~(x -2),~x,~(x+2),~are~one~ degree,~binomiamls~in~A.P.~~with~ difference~d=2.\\ They ~are ~n=4~terms.~~ G(x)= \dfrac 1 {(x-4)(x -2)x(x+2)}.\\ If~a~series~starts~at~S=6,~and~ends~at~E=12,~~Find~~\displaystyle~\sum_{x=S}^ E G(x).\\ Apart ~from~Partial~Fraction~there~is~another~method~given~below.\\ Name~~~L~~~as~Last-term,~L=(x+2) ,~~~F~as~First-term,~F=(x-4).~~K=L-F=6.\\ P(x)=\dfrac F K*G(x),~~~~Q(x)=\dfrac L K*G(x). ~~\scriptsize~ both~ expressions ~have~ only~(n-1)=3~terms~because ~of~ cancellation.\\ \displaystyle Using~ the~ formula~\large \color{#20A900}{\sum_{x=S}^E G(x)=\frac 1 K*\Bigg(\sum_{x=S}^{S+d-1} Q(x) - \sum_{x=E-d+1}^{E} P(x)\Bigg ).}\\ \displaystyle Substituting~given~values,~ \sum_{x=6}^{12} G(x)=\frac 1 6*\Big\{\sum_{x=6}^7 Q(x) - \sum_{x=11}^{12} P(x) \Big \}.\\ P(x)=\dfrac{x-4}{(x-4)(x-2)x(x+2)}=\dfrac 1 {(x-2)x(x+2)}.~~~Q(x)=\dfrac{x+2}{(x-4)(x-2)x(x+2)}=\dfrac 1 {(x-4)(x-2)x}.\\ \displaystyle \therefore~\frac 1 6*\sum_{x=6}^7 Q(x) =\frac 1 6*\Big \{\dfrac 1 {2*4*6}+\dfrac 1 {3*5*7} \Big \}=\dfrac{51}{18*560}...(a)\\ \displaystyle \therefore~\frac 1 6*\sum_{x=11}^{12} P(x) =\frac 1 6*\Big \{\dfrac 1 {9*11*13}+\dfrac 1 {10*12*14} \Big \}=\dfrac{\frac{989}{429}}{18*560}...(b)\\ \displaystyle \color{#D61F06}{ \sum_{x=6}^{12} \dfrac 1 {(x-4)(x -2)x(x+2)}=\dfrac{51 -\frac{989}{429}}{18*560}}=\dfrac{2089}{432432}\\ \text{Below we see how this formula is developed.}\\ G(x)=\dfrac 1 {(x-4)(x -2)x(x+ 2)}=\frac 1 {(x+2) - (x-4)}*\dfrac{ \color{#EC7300}{(x+2)} - \color{#20A900}{(x-4)} } { {\color{#20A900}{(x-4)}}(x-2)x \color{#EC7300}{(x+2)}}\\ =\frac 1 {(x+2) - (x-4)}*\Big \{{\color{#EC7300}{\dfrac 1{ (x-4)(x-2)x} }}- {\color{#20A900}{\dfrac 1{ (x-2)x (x+2)}}}\Big \}\\ \scriptsize \qquad \qquad \qquad \qquad ~Let ~{\color{#EC7300}{Q(x)=\dfrac 1{ (x-4)(x-2)x} }}, ~~~{\color{#20A900}{P(x)=\dfrac 1{ (x-2)x (x+2)}}} \\ \therefore~G(x)=\frac 1 {(x+2) - (x-4)}*\Big \{Q(x) -P(x)\Big \} \\ \implies~\displaystyle~ \sum_{x=6}^{12} G(x)=\frac 1 {(x+2) - (x-4)}*\Big \{\sum_{x=6}^{12} Q(x) - \sum_{x=6}^{12} P(x).\Big \} \\ \qquad \qquad \qquad \scriptsize \color{#20A900}{But~~\displaystyle \sum_{n=6}^{12}Q(x)=\sum_{n=6}^7 Q(x)+ \sum_{n=6}^{10} P(x)...Proof~in~Notes ~below.}\\ = \displaystyle \frac 1 {(x+2) - (x-4)}*\Big \{\sum_{x=6}^7 Q(x) +\sum_{x=6}^{10} P(x)- \sum_{x=8}^{10} P(x) - \sum_{11}^{12} P(x).\Big \}\\ \displaystyle \frac 1 {(x+2) - (x-4)}*\Big \{\sum_{x=6}^7 Q(x) - \sum_{11}^{12} P(x).\Big \}\\$

TO COMPLETE SOON

$\color{#20A900}{Note-~~\displaystyle \sum_{n=6}^{12}Q(x) \\ \displaystyle = \sum_{n=6}^{7} Q(x) + \sum_{n=8}^{12} Q(x)\\ \displaystyle =\sum_{n=6}^{7} Q(x)+ \sum_{n=8}^{12} \dfrac 1 {(x-4)(x -2)x(x+2)}.\\ \displaystyle =\sum_{n=6}^{7} Q(x)+ \sum_{n=6}^{10} \dfrac 1 {(x -2)x(x+2)(x+4)}.\\ \displaystyle = \sum_{n=6}^{7} Q(x)+ \sum_{n=6}^{10} P(x)}.$

Niranjan Khanderia - 4 years, 1 month ago

You're right to point out that there is a better way in certain cases like this. Partial fractions allow us to quickly reduce to a "known problem", instead of having to "Creatively determine the telescoping terms $t_k$ ".

Note: You're missing a factor of $\frac{1}{7}$ .

Calvin Lin Staff - 4 years, 1 month ago

Thanks for your comment. I have added $\frac 1 7$ .
I have changed problem so it is made more clear. Avoid two 4 in (x-4) and (x+4). Also have an upper limit so can show its working. I will add more explanation soon.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$