Checking series for convergence

In summary, checking series for convergence involves determining whether the sum of an infinite series approaches a finite limit. Various tests are used for this purpose, including the Ratio Test, Root Test, Integral Test, Comparison Test, and the p-Series Test. Each test has specific conditions and is suited for different types of series. A series is considered convergent if it approaches a finite value as more terms are added; otherwise, it is divergent. Understanding these concepts is essential in mathematical analysis and applied mathematics.
  • #1
Lambda96
226
75
Homework Statement
Check whether the series converges
Relevant Equations
All convergence criteria are allowed
Hi,

I am having problems with task d)

Bildschirmfoto 2023-12-06 um 18.30.36.png


I now wanted to check the convergence using the quotient test, so

I have now proceeded as follows:







Unfortunately I can't get any further now, if I form the limit with Mathematica, must come out, unfortunately I don't know how I can show this with my expression, or should I have used a different criterion for the task?
 
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  • #2
Look at and use
 
Last edited:
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  • #3
Lambda96 said:
Homework Statement: Check whether the series converges
Relevant Equations: All convergence criteria are allowed

Hi,

I am having problems with task d)

View attachment 336770

I now wanted to check the convergence using the quotient test, so

I have now proceeded as follows:







Unfortunately I can't get any further now, if I form the limit with Mathematica, must come out, unfortunately I don't know how I can show this with my expression, or should I have used a different criterion for the task?
Seems you could use that
 
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  • #4
WWGD said:
Seems you could use that

How does that generalize to the case where the exponent is rather than ?
 
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  • #5
Thank you pasmith and WWGD for your help 👍👍, I have now used the root test, which allowed me to reduce the expression to and which corresponds to the limit to
 
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  • #6
pasmith said:
How does that generalize to the case where the exponent is rather than ?
Well, Lambda96 used it to solve his problem. I only provide hints, as per PF policy.
 

FAQ: Checking series for convergence

What does it mean for a series to converge?

A series converges if the sequence of its partial sums approaches a finite limit as the number of terms goes to infinity. In other words, the infinite sum of the terms of the series results in a finite number.

What are some common tests for checking the convergence of a series?

Some common tests include the Ratio Test, the Root Test, the Integral Test, the Comparison Test, the Limit Comparison Test, and the Alternating Series Test. Each of these tests has specific conditions under which they can be applied to determine the convergence of a series.

How do you apply the Ratio Test to determine convergence?

To apply the Ratio Test, you take the limit of the absolute value of the ratio of successive terms in the series. If the limit is less than 1, the series converges absolutely. If the limit is greater than 1 or is infinite, the series diverges. If the limit equals 1, the test is inconclusive.

What is the difference between absolute convergence and conditional convergence?

A series is said to converge absolutely if the series of the absolute values of its terms converges. Conditional convergence occurs when a series converges, but it does not converge absolutely. In other words, the series converges only due to the specific arrangement of its terms.

Can a divergent series become convergent by rearranging its terms?

No, a divergent series cannot be made convergent by rearranging its terms. However, for conditionally convergent series, rearranging the terms can affect the sum to which the series converges or even cause it to diverge. This is known as the Riemann series theorem.

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