CEGEP Calculus II — Power Series, Taylor and Maclaurin Series Worksheet

All 7 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Power Series and the Radius of Convergence

   A ski resort models the settling of fresh snow with the power series $${\sum }_{n=0}^{\infty }\frac{(x+4{)}^n}{5^n\sqrt{n+1}}.$$ State the centre of the series, then apply the Ratio Test to determine its radius of convergence. Name the test and show the limit you evaluate.

2. Q2The Interval of Convergence

   Determine the interval of convergence of $${\sum }_{n=1}^{\infty }\frac{(x-3{)}^n}{n{4}^n}.$$ Give the centre and the radius, then test each endpoint separately, naming the test you use at each one.

3. Q3The Maclaurin Series of a Function

   Let $f(x)=\ln (1+2x)$. Working from the definition ${\displaystyle {\sum }_{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n}$ — that is, by differentiating $f$ and evaluating at $0$ — find the first four non-zero terms of the Maclaurin series of $f$. Then give the general term, and state the radius of convergence.

4. Q4Taylor Series Centred at a Point

   Find the Taylor series of $f(x)={\displaystyle \frac{1}{x}}$ centred at $a=2$, expressed with ${\displaystyle {\sum }_{n=0}^{\infty }}$ and a general term in $(x-2{)}^n$. State the centre and the radius of convergence.

5. Q5Building New Series from Known Ones

   The geometric series ${\displaystyle \frac{1}{1-t}={\sum }_{n=0}^{\infty }{t}^n}$ for $|t|<1$ is known. Without computing any Maclaurin coefficient from repeated derivatives, use substitution, multiplication and term-by-term differentiation to answer the following.

   1. Find the Maclaurin series of $g(x)={\displaystyle \frac{x}{1+9x^2}}$ and state its radius of convergence.
   2. Differentiate the geometric series term by term to obtain a series for ${\displaystyle \frac{1}{(1-t{)}^2}}$, and use it to evaluate ${\displaystyle {\sum }_{n=1}^{\infty }\frac{n}{2^{n-1}}}$.
6. Q6Evaluating an Integral with a Series

   The function $e^{-x^2}$ has no elementary antiderivative, so ${\displaystyle {\int }_0^{1/2}{e}^{-x^2}dx}$ cannot be found by the Fundamental Theorem alone.

   1. Starting from the Maclaurin series of $e^t$, write the integral as an infinite series with a general term.
   2. The series alternates. Use the Alternating Series Estimation Theorem to find how many terms guarantee an error smaller than ${10}^{-3}$, and give the value of the integral to three decimal places.
7. Q7Synthesis — drawing on several topics in this set

   This question builds a well-known series from the geometric one and then uses it numerically.

   1. Substitute into ${\displaystyle \frac{1}{1-t}={\sum }_{n=0}^{\infty }{t}^n}$ to obtain the Maclaurin series of ${\displaystyle \frac{1}{1+x^2}}$, and state its radius.
   2. Integrate term by term to obtain the Maclaurin series of $\arctan x$, justifying the value of the constant of integration.
   3. Find the interval of convergence of the series in (b), testing both endpoints separately.
   4. Use the series to approximate $\arctan \frac{1}{2}$ to three decimal places, and say how you know the error is small enough.

The 5 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The Ratio Test gives you a radius. It does not give you the interval

This is the single distinction the whole set is built around, and it is where most of the marks are lost. Q1 asks only for the centre and the radius of convergence, and the Ratio Test settles it in one limit. Q2 asks for the interval of convergence, and the same limit gets you most of the way and then stops dead.

At the two endpoints the ratio limit equals exactly 1, which is the one value the test declares itself inconclusive on. So the endpoints are not hard cases of the Ratio Test; they are outside its reach entirely, and each has to be substituted back into the original series and judged on its own merits.

Read the centre off the powers literally. The standard form is (x − a)ⁿ, so a plus sign inside the bracket is a negative centre, and Q1 opens with a plus sign for that reason. Getting the sign wrong shifts the whole interval and every endpoint with it, and nothing downstream will flag it.

Taylor and Maclaurin from the definition

Q3 and Q4 are the same machine run at two different centres, and both say to work from the definition: differentiate repeatedly, evaluate at the centre, divide by n!.

Q3 takes a logarithm at the centre 0 and asks for the first four non-zero terms, then the general term, then the radius. Q4 takes a rational function at a centre away from 0 and asks for the general term directly in powers of (x − 2). A Maclaurin series is not a different object from a Taylor series; it is the case a = 0, which is why Q4 is harder in exactly one respect — the powers stay in terms of (x − 2) throughout.

There is a free check on the radius in both questions. A Taylor series centred at a cannot converge past the nearest point where the function itself breaks down — a zero denominator, the edge of a logarithm's domain. Measure that distance from the centre and compare it with the radius your Ratio Test produced. If they disagree, one of them is wrong, and it is usually the algebra.

Build from a series you already have

Q5 forbids repeated derivatives outright: the geometric series is given, and the new series has to come out of it by substitution, multiplication and term-by-term differentiation. That prohibition is the lesson, not an obstacle. Differentiating a quotient four times to guess a pattern is slow and it is where the sign errors live; substituting into a known series is two lines and carries its own convergence condition with it. The devis expects this route, and Q6 and Q7 both depend on it.

Using a series to get a number

Q6 and Q7(d) are the payoff: an integral the Fundamental Theorem cannot reach, and a value you cannot get without a calculator, both handled by summing a few terms and bounding what you threw away.

Q7 is the whole set in one question — substitute into the geometric series, integrate term by term, justify the constant, find the interval of convergence with both endpoints tested, then use the result numerically. If you can do Q7 unaided, you can do this topic.

The tests themselves are the previous worksheet

Once an endpoint is substituted, the x disappears and you are holding a plain numerical series. Nothing in this set will tell you what to do with it — that is Sequences and Series, the previous worksheet in this course, and the p-series, comparison, limit comparison, integral and alternating series tests are all in play here. Absolute against conditional convergence matters too: at an endpoint where the series alternates, "converges" is only half an answer, because whether the convergence is conditional is part of what was asked.

Getting the most out of it

Write the centre and the radius down as named quantities

Two short lines — "centre a = …", "radius R = …" — before you go anywhere near an endpoint. Every question in this set asks for at least one of them explicitly, they are cheap marks, and having them written down is what stops the endpoint substitution going in at the wrong number.

Substitute the endpoint into the original series, not into the ratio

Put the number in, then simplify the powers before you do anything else. The geometric factor cancels against the power of the base, and what is left is nearly always a series you recognise on sight. Half the difficulty at an endpoint is that people try to test it before simplifying it into something recognisable.

Name the test, at each end

"Converges" is not an answer at a CEGEP endpoint; "converges by the Alternating Series Test, since the terms are positive, decreasing and tend to 0" is. Name the test and state the hypotheses you checked. Expect to name a different test at the other end — that is normal, not a sign that you have made a mistake.

Reach for a known series before you differentiate anything

The geometric series and the exponential series between them generate most of what this set asks for, by substitution, multiplication, differentiation or integration. Repeated differentiation is the method of last resort, and the questions that want it say so.

Sanity-check every series you produce

Expand your general term at the first two or three values of n and compare it with the terms you wrote out by hand; check the radius against the distance from the centre to the nearest place the function misbehaves; put a convenient number into a numerical sum and see whether the partial sums are heading where you claimed. These checks cost a minute and catch sign errors that nothing else will.

Common questions