Calculating Infinite Series Sum: Methods for Convergence and Divergence

In summary, the conversation revolved around finding the sum of the series \sum _{n=0} ^{\infty} \frac{2^{2n+1}x^{2n}}{n!} and whether knowledge of Taylor series is required for solving it. The potential methods discussed included using the geometric sum method, the approximation method, and finding a differential equation that the series solves. The value of the sum at x = 0 was also mentioned, as well as comparing it to the values of the listed functions at x = 0. The process of elimination was suggested, but it was noted that it may not be a useful approach in general.
  • #1
dekoi
The sum of a series:
[tex]\sum _{n=0} ^{\infty} \frac{2^{2n+1}x^{2n}}{n!}[/tex]
is:
a)[tex]2cos(2x)[/tex]
b)[tex]cos(x^2)[/tex]
c)[tex]e^{2x}[/tex]
d)[tex]2e^{2x^2}[/tex]
e) None of the above.I have absolutely no idea how I would go about solving this. I know various tests for convergence and divergence, but the only methods I have to calculate sums are the geometric sum method, and the approximation method. I'm not sure how to solve this.

Any help is greatly appreciated, Thank You.
 
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  • #2
Do you know about taylor series? Specifically, what is the taylor series for ex?
 
  • #3
No I haven't learned the Taylor Series. Is that the only way of doing this question?
 
  • #4
What's the sum of the series for x = 0? How does it compare to the values of the listed functions at
x = 0?
 
  • #5
It also might help you to note that the sum is never negative.
 
  • #6
Does this require knowledge of Talor Series?
 
  • #7
dekoi said:
Does this require knowledge of Talor Series?
No, it does not.
As Archon has pointed out in post #4, if x = 0, then what's the value of the sum? Is there any of the listed functions which returns the same value as the sum of that series at x = 0?
Can you go from here?
 
  • #8
You might be able to use the process of elimination here, but that's not a very useful approach in general. And how could you eliminate the none of the above option?

The only other way I can think of is to find a differential equation this series solves and then find the analytic solution to it. That is, defining the series as a function y(x), find an equation relating y to y' and then solve for y(x).
 
Last edited:

1. What is an infinite series?

An infinite series is a mathematical expression that represents the sum of an infinite number of terms. It is written in the form of ∑n=1∞ an, where an is the nth term of the series.

2. How do you calculate the sum of an infinite series?

The sum of an infinite series can be calculated using various methods, such as the geometric series test, the telescoping series test, and the ratio test. These methods involve analyzing the behavior of the terms in the series to determine whether the series converges (has a finite sum) or diverges (does not have a finite sum).

3. What is convergence of an infinite series?

Convergence of an infinite series refers to the behavior of the sum of the series as the number of terms increases towards infinity. If the sum approaches a finite value, the series is said to converge. If the sum does not approach a finite value, the series is said to diverge.

4. What is the difference between absolute and conditional convergence?

Absolute convergence refers to the convergence of a series when only the magnitudes of the terms are considered. Conditional convergence refers to the convergence of a series when the order of the terms is taken into account. A series that is absolutely convergent is also conditionally convergent, but the reverse is not always true.

5. Can an infinite series have more than one sum?

No, an infinite series can only have one sum. The sum of an infinite series is a unique value that represents the total of all the terms in the series. However, some series may have a sum that is undefined, such as the harmonic series, which diverges but does not have a finite sum.

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