Is the Interval of Convergence for (x-2)^n / n^(3n) from -1 to 5?

In summary: The series for x=-1 is conditionally convergent, which means that it converges but it’s not absolutely convergent. The series for x=5 is not convergent at all.In summary, the interval of convergence for the series given is from -1 to 5. Using the ratio test, it can be determined that both endpoints will result in divergence. However, the series for x=-1 is conditionally convergent, while the series for x=5 is not convergent at all. The series for x=-1 is well-known as ##\sum_{n=1}^{\infty} (-1)^n/n = - \ln 2.##
  • #1
isukatphysics69
453
8

Homework Statement


interval of convergence for
n=1 to inf
(x-2)n / n3n

Homework Equations

The Attempt at a Solution


i used the ratio test and solved for x and got that the interval of convergence is from -1 to 5. now i have to test the endpoints to determine which ones will make the series either converge or diverge. it looks to me that they both will diverge but on the multiple choice homework i do not see a choice for both diverging although i do see a choice for the -1 converging and the 5 diverging. now the 5 endpoint makes sense to me for divergence because it will be 1/n which is a divergent harmonic series
but the -1 converging is not making sense

(-3)n / n3n
(-1)1 / n
divergent

what am i not understanding here?
 
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  • #2
isukatphysics69 said:

Homework Statement


interval of convergence for
n=1 to inf
(x-2)n / n3n

Homework Equations

The Attempt at a Solution


i used the ratio test and solved for x and got that the interval of convergence is from -1 to 5. now i have to test the endpoints to determine which ones will make the series either converge or diverge. it looks to me that they both will diverge but on the multiple choice homework i do not see a choice for both diverging although i do see a choice for the -1 converging and the 5 diverging. now the 5 endpoint makes sense to me for divergence because it will be 1/n which is a divergent harmonic series
but the -1 converging is not making sense

(-3)n / n3n
(-1)1 / n
divergent

what am i not understanding here?

You are not understand that the series ##\sum 1/n## diverges but the series ##\sum (-1)^n/n## converges. (Recall the "alternating series" test.)

In fact, the second sum is well-known: ##\sum_{n=1}^{\infty} (-1)^n/n = - \ln 2.##
 
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  • #3
You don’t understand the difference between conditional and absolute convergence.
 
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Related to Is the Interval of Convergence for (x-2)^n / n^(3n) from -1 to 5?

What is an interval of convergence?

An interval of convergence is the range of values for which a given mathematical series or sequence will converge, meaning that the terms of the series approach a finite value as the number of terms increases.

How is the interval of convergence determined?

The interval of convergence is determined by the values of x for which the series converges. This can be found using various convergence tests, such as the ratio test or the root test.

What happens if the value of x is outside the interval of convergence?

If the value of x is outside the interval of convergence, the series will either diverge or the convergence cannot be determined. This means that the series either approaches infinity or oscillates between different values, rather than converging to a finite value.

Can the interval of convergence change?

Yes, the interval of convergence can change depending on the series being evaluated. For example, if a series is multiplied or divided by a constant, the interval of convergence may shift.

Why is the interval of convergence important in mathematics?

The interval of convergence is important because it determines the range of values for which a given series is valid and can be used to approximate a function. It also helps in understanding the behavior and properties of the series.

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