# For the following power series: ∑ (4^n x^n)/([log(n+1)]^(n)

• Simkate
In summary, we use the root test to determine the radius and interval of convergence for the given power series, which is found to be 0 for all values of x. Therefore, the radius of convergence is 0 and the interval of convergence is also 0. However, it is important to note that this conclusion may not be entirely accurate due to potential errors in the solution process.
Simkate
For the following power series, find

∑ (4^n x^n)/([log(n+1)]^(n)

(b) the interval of convergence, discussing the endpoint convergence when
the radius of convergence is finite.
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I wanted to know whether my solution is write, is it possible for somone to check it for me. Thank You

Due to the n-th powers, we use the root test.

r = lim(n-->∞) |4^n x^n / [log(n+1)]^n|^(1/n)
= lim(n-->∞) 4|x| / log(n+1)
= 0 for all x.

b) interval of convg=0

since the series is infinte

Are you sure you have read what the root test tells you correctly?

## 1. What is the general form of the given power series?

The general form of the given power series is ∑ (4^n x^n)/([log(n+1)]^(n)).

## 2. What is the value of x that makes the power series converge?

The value of x that makes the power series converge is -1 < x < 1.

## 3. Can the power series be used to find the sum of an infinite series?

Yes, the power series can be used to find the sum of an infinite series, but only when the value of x is within the convergence interval of -1 < x < 1.

## 4. How can the convergence of the power series be determined?

The convergence of the power series can be determined by using the ratio test, where the limit of the absolute value of the ratio of consecutive terms is taken as n approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another test must be used.

## 5. In what areas of science or mathematics is this power series commonly used?

This power series is commonly used in the fields of mathematics, physics, and engineering, particularly in the study of logarithmic functions and their applications.

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