How Do You Determine the Radius and Interval of Convergence for a Power Series?

In summary, a power series is a mathematical series that represents a function as an infinite sum of terms. The convergence of a power series can be determined using the ratio test or root test. It can only represent functions that can be written as an infinite sum of polynomials, and the radius of convergence can be found using the ratio or root test. In real-world applications, power series are commonly used in fields such as physics to approximate functions and make predictions.
  • #1
camel-man
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I need help finishing this problem I am stuck.

find radius of conv. and interval of convergence of the series. Ʃ k=0->∞ (1/k+1) (x)^k

I have found all the way up to row=1 there for it is between (0,∞) so now that means that if absolute value of x<1 it converges if >1 it diverges but I forgot how I find what x is. I thought it was R which would be 1. Someone please explain to me as elementary as possible because I am no math major this is my last course/test and I will never use this again.
 
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  • #2
camel-man said:
I need help finishing this problem I am stuck.

find radius of conv. and interval of convergence of the series. Ʃ k=0->∞ (1/k+1) (x)^k

I have found all the way up to row=1 there for it is between (0,∞) so now that means that if absolute value of x<1 it converges if >1 it diverges but I forgot how I find what x is. I thought it was R which would be 1. Someone please explain to me as elementary as possible because I am no math major this is my last course/test and I will never use this again.

We don't do test questions here.
 

FAQ: How Do You Determine the Radius and Interval of Convergence for a Power Series?

What is a power series?

A power series is a mathematical series that represents a function as an infinite sum of terms. It can be written in the form ∑n=0∞ an(x-c)n, where an are coefficients and c is a constant.

How do you determine the convergence of a power series?

The convergence of a power series can be determined by using the ratio test or the root test. If the limit of the ratio or root 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 method must be used.

Can a power series represent any function?

No, a power series can only represent functions that can be written as an infinite sum of polynomials. This includes functions such as sine, cosine, and exponential functions, but not all functions can be represented by a power series.

How do you find the radius of convergence for a power series?

The radius of convergence is the distance from the center of the series where the series converges. It can be found by using the ratio test, where the limit of the ratio of consecutive terms must be less than 1. The radius of convergence can also be found by using the root test, where the limit of the root of consecutive terms must be less than 1.

How can power series be used in real-world applications?

Power series are used in many fields of science and engineering to approximate functions and make calculations easier. They are particularly useful in physics, where they can be used to model natural phenomena and make predictions about the behavior of physical systems.

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