Recover terms or tails from partial sums

The step between two heights

The bars show the partial sums of a series, \(S_n=10(1-0.6^n)\), so bar \(n\) is as tall as the total of the first \(n\) terms.

At \(n=3\), the third bar has height 7.84 and the second bar has height 6.4. The orange bracket shows the step of 1.44 between them, which is the amount that the third term adds to the total.

Move the slider. At every \(n\), the orange step is the \(n\)th term.

The bars approach the dashed line at height 10, the sum of the series. The red bracket shows the gap from the top of bar \(n\) to this line.

This gap is the part of the series that is not yet added. At \(n=3\) it is 2.16, the sum of the fourth term and every term after it.

Terms are steps, and the tail is the gap

Move the slider. The orange step from bar \(n-1\) to bar \(n\) rises from \(S_{n-1}\) to \(S_n\), so the \(n\)th term is \(a_n=S_n-S_{n-1}\).

If the bars approach the dashed line at height \(S\), the red gap above bar \(n\) is \(S-S_n\). This gap is the tail after \(n\) terms, the sum of the terms from \(a_{n+1}\) on.

The tail starts one term after bar \(n\), because the term \(a_n\) is already inside \(S_n\).

Notation

1. Subtract two consecutive partial sums. Every term cancels except the last one.

   \[\cb{S_n}-\cb{S_{n-1}}=\cy{a_n}\]

   This holds for \(n\ge2\). For the first term, \(a_1=S_1\), because no bar stands before bar 1.

2. If the series has the sum \(S\), the tail after \(n\) terms is \(S-S_n\), which is the sum of the terms after \(a_n\). It is also the error that you make if you use \(S_n\) in place of \(S\).

   \[\begin{aligned}\cr{R_n}&=S-\cb{S_n}\\ &=a_{n+1}+a_{n+2}+a_{n+3}+\cdots\end{aligned}\]

When it fails

The rule for a term works for every series. For \(S_n=n^2\), the step from bar \(n-1\) to bar \(n\) is \(n^2-(n-1)^2=2n-1\).

The tail needs a sum \(S\) to subtract from. These bars have no limit, so there is no dashed line and the tail has no meaning.

Example 1: Find the nth term

A series has the partial sums \(S_n=n^2+2n\). Find the \(n\)th term \(a_n\), and check it for \(n=6\).

1. Write \(S_{n-1}\) by replacing \(n\) with \(n-1\) in the formula.

   \[\begin{aligned}S_{n-1}&=(n-1)^2+2(n-1)\\ &=n^2-2n+1+2n-2\\ &=n^2-1\end{aligned}\]

2. Subtract the two partial sums.

   \[\begin{aligned}a_n&=S_n-S_{n-1}\\ &=(n^2+2n)-(n^2-1)\\ &=2n+1\end{aligned}\]

3. For \(n=6\), the term is 13. As a check, \(S_6=48\) and \(S_5=35\), and \(48-35=13\).

   \[a_6=2(6)+1=13\]

4. The first term is \(a_1=S_1=3\). The formula \(2n+1\) also gives 3 at \(n=1\), so it holds for every \(n\).

Example 2: Find the tail

A series has the partial sums \(S_n=\dfrac{5n}{n+4}\). Find the tail after 6 terms, and the 7th term.

1. The sum is the limit of the partial sums.

   \[S=\lim_{n\to\infty}\frac{5n}{n+4}=5\]

2. The partial sum of 6 terms is \(S_6=\frac{30}{10}=3\).

3. A tempting wrong move: The tail after 6 terms is the sum minus the partial sum before bar 6.

   \[R_6=5-S_5=5-\frac{25}{9}=\frac{20}{9}\]

   The dashed red bracket starts at \(S_5\), below the top of bar 6, so it also counts the sixth term, which \(S_6\) already includes. The gap above bar 6 is \(5-3=2\), not about 2.22.

4. The tail after 6 terms starts at the top of bar 6, so subtract \(S_6\).

   \[R_6=5-S_6=5-3=2\]

5. The first term of the tail is the 7th term, which is the orange step from bar 6 up to bar 7.

   \[a_7=S_7-S_6=\frac{35}{11}-3=\frac{2}{11}\]

Practice

A series has the partial sums \(S_n=\dfrac{3n}{n+2}\). Find \(a_5\) and the tail after 5 terms.

On the figure, the fifth term is the step from bar 4 to bar 5. The tail is the gap from the top of bar 5 to the dashed line.

Questions

Questions 1 and 2 use the figure, which shows the first five partial sums of a series: 3, 5, 6.5, 7.5 and 8.25. The dashed line at height 9 is the sum of the series.

1. What is the fourth term, \(a_4\)?

   • 1
   • 7.5
   • 6.5
   Answer

   1

2. What is the tail after 3 terms?

   • 2.5
   • 4
   • 6.5
   Answer

   2.5

3. A series has the partial sums \(S_n=3^n-1\). What is the term \(a_5\)?

   • 162
   • 242
   • 243
   Answer

   162

Summary

You find a term by subtracting the partial sum before it, and you find a tail by subtracting the partial sum from the sum.

Each bar adds one term, so a term is the step between two bars and the tail is the gap above a bar.

You use these when a problem gives running totals and asks for one term or for what remains.

This is one of five skills in topic 10.1 of Which & Why: AP Calculus.