Introduction to Power Series: Video e Problemi di Pratica

True or False: If the power series $\sum a_n(x-3{)}^n$ converges at $x=0$ and diverges at $x=9$, then it also diverges at $x=5.9$.

Find the radius $R$ and interval of convergence for the series $\sum _{n=0}^{\infty }(-3{)}^n(n+1\left)\right(x+2{)}^n$.

Find the radius $R$ and the interval of convergence for the series $\sum _{n=1}^{\infty }\frac{(2x+1{)}^n}{3n+3}$.

Determine the interval of absolute convergence for the series $\sum _{n=0}^{\infty }(-3{)}^n(n+1\left)\right(x+2{)}^n$.

Determine the values of $x$ (if any) where the series $\sum _{n=0}^{\infty }{(-\frac{1}{2})}^n(x+2{)}^n(n+2)$ converges conditionally.

Determine the interval of absolute convergence for the series $\sum _{n=1}^{\infty }\frac{(2x+1{)}^n}{3n+3}$.

Determine the values of $x$ (if any) where the series $\sum _{n=1}^{\infty }\frac{(2x+3{)}^n}{4n+4}$ converges conditionally.

Find the radius $R$ and the interval of convergence for the series $\sum _{n=1}^{\infty }{(x-4)}^n(\sqrt{n}-\sqrt{n-1})$.

Determine the interval of absolute convergence for the series $\sum _{n=1}^{\infty }{(x-4)}^n(\sqrt{n}-\sqrt{n-1})$.

Determine the values of $x$ (if any) where the series $\sum _{n=2}^{\infty }{(x-5)}^n(\sqrt{n-1}-\sqrt{n-2})$ converges conditionally.

Let $a_1\ge a_2\ge a_3\ge \cdots$ be a nonincreasing sequence of positive numbers. Suppose the series $a_1+a_2+a_4+a_8+\cdots$ diverges. Determine whether the series $\sum _{n=1}^{\infty }\frac{a_n}{n}$ converges or diverges.

Which of the following correctly defines the sequence a_n = n^2 × (n-1)! ?

What is the value of 0! (zero factorial)?

Given the sequence b_n = n × (n+1)!, what is the value of b_2?

Simplify the expression 2^3 × 3!.

If c_n = n! and n = 4, what is c_4?

Given d_n = (n+2)! / n!, what is the value of d_3?

Why is 0! defined as 1?

Given the sequence e_n = 2^n, what is the pattern of growth for the sequence as n increases?

For the sequence f_n = n^2 × (n-2)!, calculate f_3.

Which of the following expressions could represent the nth term of a power series expansion for e^x?

What is the value of 4! ?

Given the sequence k_n = n! + n, what is the value of k_3?

Simplify the expression (3^2) × 2!.

If m_n = n^2 × (n-1)!, what is the sum of the first three terms?

Which of the following best explains why (n+1)! = (n+1) × n! is useful in simplifying sequence terms?

Using the definition 0! = 1, what is the value of 2 × 0! ?

Given the sequence p_n = n! for n = 1, 2, 3, 4, what is the pattern of growth?

For the sequence q_n = n × (n-1)!, calculate q_4.

Which of the following sequences could represent the coefficients in the Maclaurin series for sin(x)?