Infinite series, does ∑ n/2^n diverge?

In summary: The Limit Ratio Test shows that the series converges because the limit is less than 1. In summary, the series \sum^{\infty}_{n=1} \frac{n}{2^{n}} converges.
  • #1
utleysthrow
25
0

Homework Statement



[tex]\sum^{\infty}_{n=1} \frac{n}{2^{n}}[/tex]

Does this series converge or diverge?

Homework Equations





The Attempt at a Solution



By the Cauchy condensation test (http://en.wikipedia.org/wiki/Cauchy_condensation_test) I think this one diverges. But not sure if I am using it correctly.

According to the test,

[tex]\sum^{\infty}_{n=1} \frac{n}{2^{n}}[/tex]

converges if and only if

[tex]\sum^{\infty}_{n=1} 2^{n} \frac{2^{n}}{2^{n}} = \sum^{\infty}_{n=1} 2^{n}[/tex]

converges, which doesn't.

Thank you for any help.
 
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  • #2
utleysthrow said:

Homework Statement



[tex]\sum^{\infty}_{n=1} \frac{n}{2^{n}}[/tex]

Does this series converge or diverge?

Homework Equations





The Attempt at a Solution



By the Cauchy condensation test (http://en.wikipedia.org/wiki/Cauchy_condensation_test) I think this one diverges. But not sure if I am using it correctly.

According to the test,

[tex]\sum^{\infty}_{n=1} \frac{n}{2^{n}}[/tex]

converges if and only if

[tex]\sum^{\infty}_{n=1} 2^{n} \frac{2^{n}}{2^{n}} = \sum^{\infty}_{n=1} 2^{n}[/tex]

converges, which doesn't.
In the line above, you aren't using the condensation test correctly. For your series, f(n) = n/(2n), so what would be f(2n)?

A test that would be simpler to apply would be the Limit Ratio Test.
utleysthrow said:
Thank you for any help.
 
  • #3
Mark44 said:
In the line above, you aren't using the condensation test correctly. For your series, f(n) = n/(2n), so what would be f(2n)?

A test that would be simpler to apply would be the Limit Ratio Test.

Ah, okay, I see it where I went wrong..

Using the limit ratio test

[tex] lim \left| a_{n+1}/a_{n} \right| = lim \left| \frac{(n+1)/2^{n+1}}{n/2^{n}} \right| = 1/2 < 1 [/tex]

So it converges...
 
  • #4
Yes, indeed.
 

1. What is an infinite series?

An infinite series is a sum of an infinite sequence of numbers. It is represented by the symbol ∑ and can be written as ∑ an = a1 + a2 + a3 + ... + an + ....

2. What does it mean for a series to diverge?

For a series to diverge means that the sum of an infinite sequence of numbers does not have a finite value. This means that the sum either approaches infinity or alternates between different values and does not converge to a specific number.

3. How do you determine if a series diverges?

There are several tests that can be used to determine if a series diverges, such as the comparison test, the integral test, and the ratio test. These tests compare the given series to a known series with known convergence or divergence properties.

4. What is the formula for determining if ∑ n/2^n diverges?

The formula for determining if ∑ n/2^n diverges is the ratio test, which states that if the limit of |an+1/an| as n approaches infinity is greater than 1, then the series diverges. In this case, an = n/2^n, so the limit would be 1. Therefore, the series diverges.

5. Can you provide an example of a divergent series?

One example of a divergent series is the harmonic series, ∑ 1/n. This series does not have a finite sum and the terms of the series alternate between different values, leading to divergence. This can be shown using the integral test, where the integral of 1/x is ln(x), which diverges as x approaches infinity.

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