Find x for Summation Series Convergence

In summary, the task is to find the value of x for which the infinite series ( x^n/ln(n+5) ) converges. The attempt at solution involved using the ratio test and determining that |x|<1. However, it was noted that this may be overly conservative and it is necessary to consider the convergence or divergence of the series at x = +1 and x = -1. Additionally, the condition for convergence was stated as x^n/ln(n+5) approaching 0 as n approaches infinity. It was also mentioned that |x| should be strictly less than 1, not equal to.
  • #1
cloveryeah
26
0

Homework Statement



Σ ( x^n/ln(n+5) )
n=1

find the value of x that the above series converges

Homework Equations

The Attempt at a Solution


i cal. it by ratio test
and i found that |x|<1

but when i input (-1,1) into my webwork...it said it's wrong[/B]
 
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  • #2
cloveryeah said:

Homework Statement



Σ ( x^n/ln(n+5) )
n=1

find the value of x that the above series converges

Homework Equations

The Attempt at a Solution


i cal. it by ratio test
and i found that |x|<1

but when i input (-1,1) into my webwork...it said it's wrong[/B]

Your answer may be overly conservative. You need to look at whether the series converges or diverges at x = +1 and at x = -1.
 
  • #3
Hi Cloveryeah,
Great work, just something you've missed
In order for the series to converge,
Whenver n→∞ (x^n/ln(n+5)) = 0
You said that |x| < 1, you can't try x= 1 or x = -1, it's strictly less not even equal
 

1. What is the formula for finding the value of x in a summation series?

The formula for finding the value of x in a summation series is x = (n+1)/2, where n is the number of terms in the series.

2. How do you know if a summation series converges?

A summation series converges if the limit of the sequence of partial sums converges to a finite value. This can be determined by using various convergence tests, such as the ratio test or the root test.

3. Can a summation series diverge?

Yes, a summation series can diverge if the limit of the sequence of partial sums does not converge to a finite value. This can happen if the terms of the series do not approach zero or if the series oscillates between positive and negative values.

4. What is the significance of finding the value of x in a summation series?

Finding the value of x in a summation series can help determine if the series converges or diverges. It can also help in calculating the sum of the series, which is useful in various mathematical and scientific applications.

5. Are there any other methods for finding the value of x in a summation series?

Yes, there are other methods such as the integral test, comparison test, and alternating series test that can be used to determine the convergence and value of x in a summation series.

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