Sequences and Series: math test, College – download the PDF

1 Limits of sequences / 4 pts

Find the limit of each sequence, or show that it diverges.

1. \(a_n=\dfrac{4n^2-n}{2n^2+3}\)
2. \(b_n=\dfrac{(-1)^n n}{n+1}\)
3. \(c_n=\sqrt{n^2+n}-n\)

2 Geometric series / 3 pts

1. Compute \(\sum_{n=0}^{\infty}5\left(-\tfrac23\right)^n\).
2. Write \(0.4545\ldots\) as a fraction.
3. For which \(x\) does \(\sum_{n=0}^{\infty}\left(\tfrac x4\right)^n\) converge? What is its sum?

3 Choosing a test / 4 pts

Decide whether each series converges and name the test.

1. \(\sum\dfrac{5n}{n^3+2}\)
2. \(\sum\dfrac{n}{5n-1}\)
3. \(\sum\dfrac{1}{n\sqrt n}\)

4 Ratio and root tests / 4 pts

1. Use the ratio test on \(\sum\dfrac{2^n}{n!}\).
2. Use the root test on \(\sum\left(\dfrac{n}{2n+1}\right)^n\).

5 An alternating series / 3 pts

Study \(\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{2n+1}\): does it converge absolutely, conditionally, or diverge?

6 Power and Taylor series / 2 pts

1. Write the Maclaurin series of \(\dfrac{1}{1+x^2}\) using \(\dfrac{1}{1-u}=\sum u^n\).
2. What is its radius of convergence?