Sequences and series convergence – university Year 1 maths worksheet with answers to download as PDF

Find the limit of each sequence as \(n\to\infty\), showing your method.

1. \(a_n=\dfrac{5n^2-3n}{2n^2+7}\) [2 marks]
2. \(b_n=\sqrt{n^2+n}-n\) [3 marks]
3. \(c_n=\left(1+\dfrac2n\right)^n\) [2 marks]
4. \(d_n=n^{1/n}\) [3 marks]

Let \(a_n=\dfrac{3n+1}{n+2}\).

1. Show that \(|a_n-3|=\dfrac5{n+2}\). [2 marks]
2. Find the smallest \(N\) such that \(|a_n-3|<0.01\) for all \(n>N\). [2 marks]
3. Prove from the definition that \(a_n\to3\). [3 marks]

Find the sum of each series, or explain why it diverges.

1. \(\displaystyle\sum_{n=0}^\infty5\left(\tfrac23\right)^n\) [2 marks]
2. \(\displaystyle\sum_{n=1}^\infty\dfrac1{n(n+2)}\) [3 marks]
3. \(\displaystyle\sum_{n=1}^\infty(0.9)^n\) [2 marks]
4. Write \(0.\overline{27}=0.272727\ldots\) as a fraction. [2 marks]

Decide whether each series converges, naming the test and the series you compare with.

1. \(\sum\dfrac1{n^2+3n}\) [2 marks]
2. \(\sum\dfrac1{\sqrt n+1}\) [2 marks]
3. \(\sum\dfrac{n+1}{n^3+2}\) [2 marks]
4. \(\sum\dfrac{\ln n}n\) [2 marks]

Use the ratio test or the root test.

1. \(\sum\dfrac{3^n}{n!}\) [2 marks]
2. \(\sum\dfrac{n^2}{2^n}\) [2 marks]
3. \(\sum\dfrac{n!}{2^n}\) [2 marks]
4. \(\sum\left(\dfrac{2n}{3n+1}\right)^n\) [2 marks]

Answer each part.

1. Show that \(\sum\dfrac{(-1)^n}{\sqrt n}\) converges but not absolutely. [3 marks]
2. Is \(\sum\dfrac{(-1)^nn}{n^2+1}\) absolutely convergent, conditionally convergent or divergent? [3 marks]
3. Find the radius and interval of convergence of \(\sum_{n\ge1}\dfrac{n\,x^n}{3^n}\). [4 marks]

A ball is dropped from a height of 2 m. After each bounce it rises to \(\frac34\) of its previous height. A fall from height \(h\) takes \(\sqrt{2h/g}\) seconds, with \(g=9.8\) m s\(^{-2}\).

1. Complete the table.
2. Find the total distance travelled by the ball, assuming it bounces for ever.
3. Show that the times of successive falls form a geometric sequence with ratio \(\sqrt{0.75}\).
4. Find the total time before the ball comes to rest.
5. Comment on how a series can have infinitely many terms but a finite total time.

A patient takes 100 mg of a medicine every 12 hours. After 12 hours only 30% of the amount in the body remains. Let \(a_n\) be the amount just after the \(n\)th dose.

1. Show that \(a_1=100\), \(a_2=130\), \(a_3=139\).
2. Show that \(a_n=100(1+0.3+0.3^2+\dots+0.3^{n-1})\) and find a closed form.
3. Find \(a_5\) and the long-term level \(\lim a_n\).
4. The safe maximum level is 150 mg. Is the dose safe in the long run? What dose would give a long-term level of 120 mg?

Priya pays £200 into a savings account at the end of every month. The account pays 0.4% interest per month, added at the end of each month before her payment.

1. Show that after 3 months she has \(200(1.004^2+1.004+1)\).
2. Show that after \(n\) months the balance is \(B_n=200\dfrac{1.004^n-1}{0.004}\).
3. Find the balance after 24 months and the interest earned.
4. How many months does it take for the balance to exceed £10 000?

Achilles runs at 10 m/s and a tortoise at 1 m/s. The tortoise starts 100 m ahead. Zeno argued: when Achilles reaches the tortoise's starting point, it has moved on; when he reaches that point, it has moved on again, and so on.

1. Explain the times in the table and complete it.
2. Find the total time by summing the geometric series.
3. Check by solving \(10t=100+t\) directly.
4. How far does Achilles run before he catches the tortoise?

Start with an equilateral triangle of side 1 and area \(A_0\). At each stage, every side is divided into three and an outward equilateral triangle is built on the middle third.

1. Show that the perimeter at stage \(n\) is \(3\left(\tfrac43\right)^n\) and that it tends to infinity.
2. At stage \(n\ge1\), \(3\times4^{n-1}\) new triangles are added, each with area \(A_0/9^n\). Write the total added area as a series.
3. Show that the area tends to \(\frac85A_0\).
4. Comment on a shape with infinite perimeter but finite area.

Identical books of length 1 are stacked at the edge of a table. With the best arrangement, the top book can stick out by \(\frac12\), the next by \(\frac14\) more, then \(\frac16\), and so on: with \(n\) books the overhang is \(\dfrac12H_n\), where \(H_n=1+\frac12+\dots+\frac1n\).

1. Complete the table.
2. How many books are needed for the top book to be completely beyond the table edge?
3. Explain why any overhang, however large, is possible in theory.
4. It takes 31 books to reach an overhang of 2 lengths. Check with \(H_{30}\approx3.995\) and \(H_{31}\approx4.027\).

A calculator evaluates \(e^{0.5}\) using partial sums \(S_N=\displaystyle\sum_{k=0}^N\dfrac{0.5^k}{k!}\).

1. Find \(S_6\) to 7 decimal places.
2. Show that the tail after \(S_N\) is less than \(\dfrac{0.5^{N+1}}{(N+1)!}\cdot\dfrac1{1-0.5}\).
3. Find the smallest \(N\) that guarantees an error below \(10^{-6}\).
4. Why does this series converge for every \(x\)?

A lake is restocked every year. Each year 20% of the fish die or are caught, and then 400 new fish are added: \(a_{n+1}=0.8a_n+400\), with \(a_0=500\).

1. Find \(a_1\), \(a_2\), \(a_3\).
2. If the sequence converges to \(L\), show that \(L=2000\).
3. Show that \(a_n=2000-1500(0.8)^n\).
4. Prove that \((a_n)\) is increasing and bounded above by 2000.
5. After how many years does the population first exceed 1900?

Euler proved in 1734 that \(\displaystyle\sum_{n=1}^\infty\dfrac1{n^2}=\dfrac{\pi^2}6\). Before that, mathematicians could only estimate it. Let \(S_N=\displaystyle\sum_{n=1}^N\dfrac1{n^2}\) and \(S\) the full sum.

1. Use the integral test to show \(\displaystyle\int_{N+1}^\infty\dfrac{dx}{x^2}\lt S-S_N\lt \int_N^\infty\dfrac{dx}{x^2}\).
2. Deduce that \(S_N+\dfrac1{N+1}\lt S\lt S_N+\dfrac1N\).
3. Use \(N=10\) to find bounds for \(S\).
4. Compare with \(\pi^2/6\) and say how many terms would be needed to get the same accuracy from \(S_N\) alone.

Start from \(\dfrac1{1-x}=\displaystyle\sum_{n\ge0}x^n\) for \(|x|<1\).

1. Integrate term by term to show \(-\ln(1-x)=\displaystyle\sum_{n\ge1}\dfrac{x^n}n\) for \(|x|<1\).
2. Replace \(x\) by \(-x\) and, admitting that it holds at \(x=1\), deduce \(\ln2=1-\frac12+\frac13-\frac14+\cdots\).
3. Using the alternating series bound, how many terms of this series guarantee an error below \(0.001\)?
4. Put \(x=\frac12\) in part (a) to get \(\ln2=\displaystyle\sum_{n\ge1}\dfrac1{n2^n}\). Show that 10 terms give an error below \(10^{-4}\).
5. Which series would you program, and why?

A fair die is rolled until the first six. The number of rolls \(X\) has \(P(X=n)=\frac16\left(\frac56\right)^{n-1}\), so \(E[X]=\displaystyle\sum_{n\ge1}n\,\tfrac16\left(\tfrac56\right)^{n-1}\).

1. Complete the table and check that the probabilities add up to 1.
2. Use the ratio test to show that the series for \(E[X]\) converges.
3. Differentiate \(\sum x^n=\frac1{1-x}\) to show \(\sum nx^{n-1}=\dfrac1{(1-x)^2}\) for \(|x|<1\).
4. Deduce \(E[X]\).
5. Find \(P(X>12)\) and interpret.

The alternating harmonic series \(1-\frac12+\frac13-\frac14+\cdots\) converges to \(\ln2\approx0.6931\). Now rearrange the same terms: one positive term followed by two negative terms.

1. Show that the \(k\)th group equals \(\dfrac1{2(2k-1)}-\dfrac1{4k}\).
2. Deduce that the rearranged series has sum \(\frac12\ln2\).
3. Explain why this does not contradict the original sum.
4. What does Riemann's rearrangement theorem say, and why can it not happen for \(\sum\frac{(-1)^n}{n^2}\)?

The integral \(I=\displaystyle\int_0^1e^{-x^2}dx\) appears in probability, but \(e^{-x^2}\) has no elementary antiderivative. Engineers use a series instead.

1. Write the Maclaurin series of \(e^{-x^2}\) and state its radius of convergence.
2. Integrate term by term to show \(I=\displaystyle\sum_{n\ge0}\dfrac{(-1)^n}{n!(2n+1)}\).
3. Compute the sum of the first five terms.
4. Use the alternating series bound to estimate the error.
5. The true value is \(0.746824\). Compare.

We know \(e=\displaystyle\sum_{k=0}^\infty\dfrac1{k!}\). Suppose, for a contradiction, that \(e=\dfrac ab\) with positive integers \(a,b\).

1. Show that for any \(n\), \(0\lt e-\displaystyle\sum_{k=0}^n\dfrac1{k!}\lt \dfrac1{n!\,n}\) by comparing the tail with a geometric series.
2. Let \(x=b!\left(e-\displaystyle\sum_{k=0}^b\dfrac1{k!}\right)\). Show that \(x\) is an integer.
3. Show that \(0\lt x\lt \dfrac1b\le1\).
4. Conclude.