Is Power Series Convergence Related to Other Series Convergence?

In summary, if the series \sum_{n=0}^{\infty} c_{n}4^n is convergent, this does not necessarily mean that the series \sum_{n=0}^{\infty} c_{n}(-2)^n or \sum_{n=0}^{\infty} c_{n}(-4)^n are also convergent. The ratio test and alternating series test can be used to determine the convergence of these series.
  • #1
Brilliant
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0

Homework Statement


If [tex]\sum_{n=0}^{\infty} c_{n}4^n[/tex] is convergent, does it follow that the following series are convergent?

a) [tex]\sum_{n=0}^{\infty} c_{n}(-2)^n[/tex] b) [tex]\sum_{n=0}^{\infty} c_{n}(-4)^n[/tex]


Homework Equations


The Power Series: [tex]\sum_{n=0}^{\infty} c_{n}(x - a)^n[/tex]


The Attempt at a Solution


I was able to work all the problems that asked me to solve for a radius of convergence, but this question seems much different, and I can't think about how to prove or disprove either a or b. Any tips would be much appreciated.
 
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  • #2
If you know that:
[tex]
\sum_{n=0}^{\infty} c_{n}4^n
[/tex]
Then apply the ratio test on this to get a relationship between c_{n} and c_{n+1}, then you can use this to check the other series . Look up the alternating series test also.
 

FAQ: Is Power Series Convergence Related to Other Series Convergence?

1. What is a power series?

A power series is an infinite series of the form ∑n=0 cnxn, where cn are constants, x is the variable, and n is the power. It is a type of mathematical series that is used to represent functions as a sum of terms with increasing powers of x.

2. What does it mean for a power series to converge?

A power series converges if the sum of its terms approaches a finite value as the number of terms increases. In other words, as n approaches infinity, the value of the sum, ∑n=0 cnxn, approaches a finite limit. If this is not the case, the power series is said to diverge.

3. 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. The ratio test compares the absolute value of the ratio of consecutive terms to a limit, while the root test compares the absolute value of the nth root of the terms to a limit. If the limit is less than 1, the series converges; if it is greater than 1, the series diverges.

4. What is the radius of convergence for a power series?

The radius of convergence, denoted by R, is the distance from the center of a power series to the point at which it converges. It can be calculated using the ratio test or the root test. If the series converges for all values of x within a certain radius, the radius of convergence is infinite. Otherwise, it is a finite positive value.

5. Can a power series converge at its endpoints?

No, a power series can only converge at its endpoints if the series is a geometric series, meaning the ratio of consecutive terms is constant. In all other cases, the power series will either converge or diverge within the interval between the endpoints, with the endpoints themselves being points of divergence.

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