Extremely simple question regarding the convergence of a series

In summary, the conversation discusses an alternating series problem involving the sum of (-1)^j times the square root of j divided by j+5. While the limit of the series approaches zero, the convergence of the non-alternating series is still uncertain. The conversation suggests using the fact that for positive values, x^2 > y^2 if and only if x > y, and a/b > m/n if and only if an > bm, to analyze the algebraic expressions involved and determine the convergence of the series.
  • #1
TheFerruccio
220
0
First off, this is not a homework question. I am helping someone with an alternating series, and, for some reason, I am finding myself completely baffled by this one.

I have an alternating series that takes the form:

[itex]\sum\limits_{j=1}^\infty(-1)^j\frac{\sqrt{j}}{j+5}[/itex]

I know that the limit approaches zero, so the first requirement of the alternating series test is filled.

However, I have been doing all sorts of algebra, whether it's fractions, differences, comparing two fractions, in order to figure out whether the non alternating series converges always, meaning, will each subsequent term always be less than the previous term.

I have not been able to figure this part out. I keep ending up with funky comparisons that are not mathematically rigorous. How would you go about doing the algebra for this problem? Is there another approach that can be used?
 
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  • #2
Use the fact that, for x and y positive, [itex]x^2> y^2[/itex] if and only if x> y.
[itex]a_j^2= \frac{j}{(j+5)^2}[/itex] and [itex]a_{j+1}^2= \frac{j+1}{(j+6)^2}[/itex].

Now use the fact that, for a, b, m, n all positive, a/b> m/n if and only if an> bm.
 
  • #3
For j large, the terms are ≈ 1/√j , so the series of absolute values diverges.
 

1. What does it mean for a series to converge?

Convergence of a series means that the sum of its terms approaches a finite value as the number of terms increases. In simpler terms, the series has a finite and definite sum.

2. How can I determine if a series converges or diverges?

There are several methods to determine convergence or divergence of a series, such as the comparison test, the integral test, and the ratio test. These tests involve comparing the given series to known series with known convergence or divergence behavior.

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

Absolute convergence means that the series converges regardless of the order in which the terms are added. Conditional convergence means that the series only converges if the terms are added in a specific order.

4. Can a series converge to a negative or complex number?

Yes, a series can converge to a negative or complex number. The convergence of a series depends on the behavior of the terms, not the actual value of the sum. So, as long as the sum approaches a finite value, the series is considered to be convergent.

5. Is it possible for a series to converge and diverge at the same time?

No, a series cannot converge and diverge at the same time. A series will either converge or diverge, depending on the behavior of its terms. It cannot exhibit both behaviors simultaneously.

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