What if there exists a real number R > 0 such that a power series centered at x=a converges for |x−a| < R and diverges for |x−a| > R, then R is ______.

What is the radius of convergence when the power series only converges at x=a?

What is the radius of convergence if the power series

converges for all real numbers x?

What is the sigma notation for this expression as an infinite series $$1\ +\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\ +...$$  ?

What is the sigma notation for this expression as an infinite series $$1\ -\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\ +...$$  ?

Determine whether the series converges or diverges, $$\sum_{n=1}^{\infty}\frac{n}{n+2}$$  .

Determine whether the series converges or diverges, $$\sum_{n=1}^{\infty}\frac{n}{n+1000}$$  .

Determine whether the series converges or diverges, $$1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+...$$  .

If the sequenceof partial sums converges to a real number S, do the infinite series

converges or diverges?

This technique is important because it is used to prove the divergence or convergence of many other series. This test compares an infinite sum to an improper integral. It is important to note that this test can only be applied when we are considering a series whose terms are all positive. This test is called ______.

For any real number p, the series $$\sum_{n=1}^{\infty}\frac{1}{n_p}$$  . What is this called?

Determine $$\sum_{n=1}^{\infty}=\frac{1}{n^4}$$  whether it converges or diverges.

Determine $$\sum_{n=1}^{\infty}=\frac{1}{n^{\frac{2}{3}}}$$  whether it converges or diverges.

Determine $$\sum_{n=1}^{\infty}=\frac{1}{n^{\frac{5}{4}}}$$  whether it converges or diverges.

What is that for a series $$\sum_{n=1}^{\infty}1a_n$$  with positive terms $$a_n$$ and a continuous, decreasing function $$f$$  such that $$f\left(n\right)=a_n$$  for all positive integers n the _____________ $$R_N=\sum_{n=1}^{\infty}a_n-\sum_{n-1}^Na_{n\ }$$  satisfies the following: $$\int_{n+1}^{\infty}f\left(x\right)dx<R_N<\int_N^{\infty}f\left(x\right)dx$$  ?

When using the comparison tests, a series $$\sum_{n=1}^{\infty}a_n$$  is often compared to a ________.

A series whose terms alternate between positive and negative values is _______.

What is that if the series $$\sum_{n=1}^{\infty}\left|a_n\right|$$  converges, the series $$\sum_{n=1}^{\infty}a_n$$  is said to?

What is that if the series $$\sum_{n=1}^{\infty}a_n$$  converges, but the series $$\sum_{n=1}^{\infty}\left|a_n\right|$$  diverges, the series $$\sum_{n=1}^{\infty}a_n$$  is said to?

What is that for any series $$\sum_{n=1}^{\infty}a_n$$  , let p = $$\lim_{n\rightarrow\infty}\sqrt[n]{\left|a_n\right|}$$  ?

What is that for any series $$\sum_{n=1}^{\infty}a_n$$  , let p = $$\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|$$  ?

Using the root test, determine whether the series converges or diverges: $$\sum_{n=1}^{\infty}\frac{\left(n^2+3n\right)^n}{\left(4n^2+5\right)^n}$$  .

For the series, $$\sum_{n=1}^{\infty}\frac{2^n}{n!}$$  , using ratio test to determine whether the series converges or diverges. What is its p value?

What is it that it can be reindexed to be written in the form a+ar+ar²+⋯, where a is the initial termand r is the ratio?

What is the test that cannot prove the convergence of a series?

What is that for p = 1, we have the harmonic series $$\sum_{n=1}^{\infty}\frac{1}{n}$$  ?

What is the test that is often used for series involving factorials or exponentials?

What is the test that is for $$\sum_{n=1}^{\infty}a_n$$  with nonnegative terms, compare with a known series $$\sum_{n=1}^{\infty}b_n$$  ?

What is the series called that can be used to help approximate integrals that cannot be evaluated by other means?

What is for any real number r, the Maclaurin series $$f\left(x\right)=\left(1+x\right)^r$$  is known as?