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How wrong a truncated alternating series can be: the size of the first term left out, the side its sign points to, and the same question run backwards to find how many terms a stated accuracy needs.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you can check the two conditions the alternating series bound needs — signs that alternate and sizes that decrease to zero — and then say how far a truncated sum can be from the true one, using the size of the first term left out and the side its sign points to. You can also run the question backwards, finding how many terms a stated accuracy requires, and say why a bound that is too small is worse than no bound at all.
You can decide whether a series converges. That is a statement about the whole infinite sum, and it says nothing at all about how close the first few terms get — which is the only thing anyone can actually compute. This lesson answers the second question for the easiest case there is, and the next one answers it in general.
Truncation: stopping a series after finitely many terms.
Partial sum $s_k$: what you get when you stop after $k$ terms.
Remainder: the exact difference between the true sum and $s_k$ — a number nobody knows, or there would be nothing to approximate.
Bound: a number the remainder is guaranteed not to exceed. Not the error: an upper limit on it.
The theorem. If the terms of a series alternate in sign and their sizes decrease to zero, then after any number of terms
$$|\text{error}| \le \text{the size of the first term left out}$$
and the true sum lies on the side that omitted term points. You get a direction as well as a size, which is more than any other bound in the course gives you.
Why it is true is visible rather than technical. Each new term is smaller than the last and has the opposite sign, so the partial sums overshoot, undershoot, overshoot — each time by less. The true sum is therefore always trapped between two consecutive partial sums, and the gap between them is exactly the next term.
Both conditions are needed. Alternating alone is not enough: $1 - 1 + 1 - 1 + \cdots$ alternates and has no sum. Decreasing to zero alone is not enough either — that is the harmonic series. The bound is a theorem, and a theorem quoted without its conditions has not been applied.
Running it backwards. In practice nobody asks how wrong $10$ terms are; they say how accurate the answer must be and ask how many terms that takes. Write the bound with the number of terms unknown, set it below the accuracy required, and solve. That step is algebra, not calculus.
Another way: picture
Picture someone walking to a wall, turning, walking back a shorter distance, turning again, and so on, each leg shorter than the last. They are never at the limit point, but after any turn they are certainly within one leg of it — and the leg they are about to walk is the first term left out. That is the whole theorem, standing up.
Another way: steps
| Accuracy wanted | Terms of $1 - \tfrac12 + \tfrac13 - \cdots$ | Terms of $1 - \tfrac12 + \tfrac14 - \cdots$ |
|---|---|---|
| $0.1$ | 9 | 4 |
| $0.01$ | 99 | 7 |
| $0.001$ | 999 | 10 |
Both series converge, and one of them is useless. Each extra decimal place costs the first series ten times as many terms and the second about three more. Convergence and usefulness are different properties, and the bound is what tells them apart.
Bounding with the last term included. The bound is the first term left out. Off by one and the claimed accuracy is better than the truth, which is the one direction of error that matters: a bound that is too generous is merely unhelpful, a bound that is too tight is wrong.
Quoting the bound where the terms do not shrink to zero. Then the series does not converge and there is nothing to be close to.
Reporting the number without the bound. An approximation reported without a bound is not a result. The bound is what turns a number into a statement: the true value lies within this distance of the one computed.
$1 - \tfrac12 + \tfrac13 - \tfrac14$ keeps four terms, and the terms alternate and shrink to zero.
The conditions hold.
The first term left out is $\tfrac15$, so the error is below $0.2$.
Term five.
That term is positive, so the true sum is above the partial sum. The answer is $0.5833$ give or take $0.2$, and above it.
$1 - \tfrac12 + \tfrac14 - \tfrac18 + \cdots$ alternates and halves each time.
Geometric, ratio $-\tfrac12$.
After ten terms the first one left out is $\tfrac{1}{1024}$, so the error is below a thousandth.
Ten terms, three places.
The alternating harmonic series needs about a thousand terms for the same accuracy. How fast the terms shrink is everything.
The error after $k$ terms is at most $\dfrac{1}{k + 1}$, since the terms are the reciprocals.
Write the bound with $k$ unknown.
Requiring $\dfrac{1}{k + 1} \le 0.01$ means $k + 1 \ge 100$.
Reciprocals turn the inequality round.
So $99$ terms — for two decimal places. Convergence is not the same as usefulness, and the bound is how you find out which you have.
The series $9 - 9/2 + 9/4 - \cdots$ halves in size each time and alternates. Its partial sums after two and after three terms are $9/2$ and $27/4$. Mark its true sum on the line.
0 |——————————| 9
Mark the position with a cross, then write the value:
The alternating series $1 - \dfrac{1}{2} + \dfrac{1}{3} - \cdots$ is cut off after each of three numbers of terms. Give the largest the error can be in each case.
| Largest possible error | |
|---|---|
| Cut off after $4$ terms | |
| Cut off after $5$ terms | |
| Cut off after $6$ terms |
Put the steps of bounding the error after $5$ terms of an alternating series into the order you do them.
Number the steps in order (write the number in the box):
The alternating series $1 - \dfrac{1}{2} + \dfrac{1}{3} - \cdots$ is truncated after $8$ terms. What is the largest the error can be?
Answer:
An alternating series is truncated after $3$ terms. Which term bounds the error?
The series $1 - \dfrac{1}{2} + \dfrac{1}{3} - \cdots$ is to be summed to within $\dfrac{1}{30}$. Give the smallest value $k + 1$ may take, then the number of terms $k$ needed.
smallest $k + 1$: p, terms needed $k$: q
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
The series $9 - 9/2 + 9/4 - \cdots$ halves in size each time and alternates. Its partial sums after two and after three terms are $9/2$ and $27/4$. Mark its true sum on the line.
0 |——————————| 9
Mark the position with a cross, then write the value:
You can bound the error of an alternating series and say which side the true sum is on. Without looking: which term gives the bound, what two conditions have to hold before you may quote it, and why is a bound that is too small worse than one that is too large?
9. Your turn: how many terms of $1 - \tfrac12 + \tfrac13 - \cdots$ give an error below $0.01$?, step 3