Converging Series: Tests & Tips for Finding Solutions

In summary, the individual is seeking assistance in determining whether the three given series converge. They have attempted to use the AHS test for the first series, but were unable to show that it decreases. They have also struggled with finding integrals for the integral test and the ratio test did not work. They are requesting guidance and help with these series.
  • #1
marky1
1
0
Hi,

I would like to as you you help please with finding whether the following three series converge.

\sum_{1}^{\infty} (-1)kk3(5+k)-2k

$$\sum_{k=1}^\infty(-1)^kk^3(5+k)^{-2k}$$

\sum_{2}^{\infty} sin(Pi/2+kPi)/(k0.5lnk)

$$\sum_{k=2}^\infty\frac{\sin\left(\frac{\pi}{2}+k\pi\right)}{\sqrt k\ln k}$$

\sum_{1}^{\infty} (ksin(1+3)/(k+lnk)

$$\sum_{k=1}^\infty\frac{k\sin(1+?)}{k+\ln k}$$

I would be very grateful should you like to give me some hint (e.g. which test I should use), please.

For instance, for the first one I have tried the AHS test, but failed in showing that the series decreases.

For the second and third ones, I have not been able to find the integrals for the integral test and the ratio test seemed not to work either. I'm quite desperate, honestly.

Many thanks for any pointer and help.
 
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  • #2
Hi marky and welcome to MHB! :D

Notice that I've edited your post to include your sums in proper $\LaTeX$ code. I've left the originals so if there are any discrepancies you can point them out. Also, the third sum contains an unmatched exponent, so clarification is needed.

It's probably best that you show your work so that we may point out any errors and give guidance where needed instead of merely posting the methods, which may differ from what you have already learned and/or deduced.

Thanks.
 

1. What is the purpose of studying converging series?

The purpose of studying converging series is to understand how to determine whether a series will converge or diverge, and if it converges, what value it converges to. This knowledge is important in various fields of science and mathematics, such as physics, engineering, and statistics.

2. What are some common tests used to determine convergence of a series?

Some common tests used to determine convergence of a series include the ratio test, the root test, the integral test, and the comparison test. These tests involve comparing the given series to a known converging or diverging series.

3. How do these tests work?

These tests work by evaluating the behavior of the terms in the series as n approaches infinity. The ratio test and root test use the ratio or root of consecutive terms to determine convergence, while the integral test compares the series to an improper integral. The comparison test compares the given series to a known converging or diverging series.

4. What are some tips for finding solutions to converging series?

Some tips for finding solutions to converging series include recognizing patterns and applying known tests, breaking the series into smaller parts that are easier to evaluate, and using algebraic manipulations to simplify the series. It is also important to carefully check the conditions of the tests being used.

5. What are some real-world applications of converging series?

Converging series have many real-world applications, such as determining the behavior of electrical circuits, predicting future populations in biology and economics, and analyzing the stability of structures in engineering. They are also used in statistics to calculate the probability of certain events occurring.

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