# Find the interval & radius of convergence for the power series in #1b?

interval:

radius:

use the ratio test:

[note: for the series

-is

-is

-is

if you apply the ratio test to this:

simplifying the limit:

[note: here you can divide both the numerator and denominator by

back to the ratio test, the series can only converge if

Case 1:

Case 2:

if

this is the alternating harmonic series, which converges by the alternating series test

if

this is the harmonic series, which diverges. here is a proof

so include

radius of convergence is half the difference between the upper and lower values for the interval

and here is a video with a similar problem

By signing up, you agree to our Terms of Service and Privacy Policy

To find the interval and radius of convergence for the power series in problem 1b, you can use the Ratio Test. The Ratio Test states that for a power series ( \sum_{n=0}^{\infty} a_n(x - c)^n ), the series converges absolutely if the limit ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ) exists and is less than 1.

Apply the Ratio Test to the given power series to find the interval and radius of convergence. This involves finding the limit:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}(x - c)^{n+1}}{a_n(x - c)^n} \right| ]

Simplify the expression and solve for ( x ) to determine the interval of convergence. The radius of convergence is then the distance from the center ( c ) to the nearest point where the series converges.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the interval of convergence of #sum [(-1)^n(2^n)(x^n)] #?
- How do you find the Maclaurin Series for #f(x)=sin (2x) #?
- What is the Taylor Series generated by #f(x) = x - x^3#, centered around a = -2?
- How do you find the maclaurin series expansion of #sin^3 (x)#?
- How do you find the Maclaurin series for #ln((1+x)/(1-x))#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7