Sequence and Series

5: Determine whether the alternating series converges.

\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{2}}

Sequence and Series

Example 5: Show that the ratio test is inconclusive for the following series. Then use another method to determine the convergence or divergences of the series.

\sum_{n=1}^{\infty} \frac{n^{2}+1}{(n+1)^{2}}

Sequence and Series

\sum_{n=1}^{\infty} \frac{n^{2}-4}{2^{n}}

Use the ratio test to determine whether the series converges or diverges.

Sequence and Series

Example 3: Use any test covered in 11.1-11.6 to determine whether the following seriesconverges ordiverges.

\sum_{n=1}^{\infty} \frac{1-\sin n}{n^{3 / 2}}

Sequence and Series

Example 2: Use the limit comparison test to determine whether the following series converges or diverges.

\sum_{n=1}^{\infty} \frac{n^{2}-2 n+5}{\sqrt{3+n^{5}}}

Sequence and Series

Example 1: Use the comparison test to determine whether the following series converges or diverges.

\sum_{n=1}^{\infty} \frac{\ln n}{n}

Sequence and Series

Determine the Taylor series of the given function. Find the radius of convergence.

f(x)=\cos \left(2 x^{3}\right)

Sequence and Series

Find the power series for the given function. What is the radius of convergence?

f(x)=\frac{1}{1+3 x}

Sequence and Series

Determine the interval of convergence for the given power series.

\sum_{n=0}^{\infty} \frac{(3 x)^{n}}{n !}

Sequence and Series

5: Decide whether the series converges. If so, find its sum.

\sum_{n=1}^{\infty} \frac{3}{n(n+1)}

\sum_{n=2}^{\infty} \frac{3^{n}-1}{5^{n-2}}

Use the Ratio or Root test, to determine whether the series converges or diverges.

\sum_{n=1}^{\infty} \frac{(\ln n)^{2 n}}{n^{n+1}}