Solving Convergence Tests: Tips & Tricks

[matrix_204]

Convergence tests?

I was having some trouble deciding which convergence tests to use for some of the following problems, as i have about a day or less to work on them. So please just tell me which convergence tests are easiest in doing these problems and some tips as sum of them require more than normal time to do.

1)a) let [S_k]_k>=1 be a sequnce with positive integers in increasing order and that do not have a 6 in their decimal expansion. For example, the sequence begins as; 1,2,3,4,5,6,7,8,...,15,17,...59,70,...
Prove carefully that the (sum of the series 1/N_k as k goes from 1 to infinity) converges to a number less than 82.

b) Now for the other part, make a sequence(say {M_k}_k>=1) such that it contains positive integers in increasing order and whose decimal expansions end in 6.
Now prove that this series diverges. (i.e. sum of 1/M_k as k goes from 1 to infinity)


2. Let a, b >=0, prove that the lim of (a^n + b^n)^1/n +max{a,b} as n goes to infinity.(this one doesn't seem very difficult and i have yet to do this one, but not much to worry about)

3.i) This problem i kinda had an idea but i think i lost it, i had this problem given long ago, but it goes like this, prove that if 0<a<2 then a<root2a<2. (this part of the problem doesn't seem to relate to convergence but i guess it leads to the second part).

ii) Prove the sequence {root2, root(2root2),root(2root(2root2)),...} converges. And what is its limit?
(so what i think u have to do here is, define a sequence, say {a_n}_n>=0 recursively (btw would someone help me understand how you define recursive formulas, i mean I've done some problems where u are given certain sequences but i don't kno how to come up with a formula, so if anyone knows an easy way to make me understand this, please mention it).
let a_0=root2 and a_n+1=root(2(a_n)), then a_n converges.)

 

[Nate810]

For the first two problems, use the ratio test or the root test for absolute convergence. For the third problem, you can use the Monotone Convergence Theorem to prove that the sequence {a_n} converges.

 

[thepatientmental]

In general, when faced with a convergence test problem, it can be helpful to follow these tips and tricks:

1. Identify the type of series: Before applying any convergence tests, it's important to identify the type of series you are dealing with. Is it a geometric series, a p-series, an alternating series, etc.? This will help guide your approach and determine which convergence test is most appropriate.

2. Use the comparison test: The comparison test is a useful tool for determining the convergence or divergence of a series. If you have a series that is difficult to evaluate, try comparing it to a simpler series whose convergence is already known.

3. Be familiar with the convergence tests: Make sure you are familiar with the various convergence tests, such as the ratio test, root test, integral test, etc. This will help you choose the most efficient test for a particular series.

4. Look for patterns: In some cases, you may be able to spot patterns in the series that can help you determine convergence or divergence. For example, in the first problem, you can see that the sum of the series is bounded above by 1 + 1/2 + 1/4 + 1/8 + ... which is a geometric series with a sum of 2.

5. Use algebraic manipulation: Sometimes, manipulating the terms of a series algebraically can help you determine convergence. For example, in the second problem, you can rewrite (a^n + b^n)^1/n as (a^n(1 + (b/a)^n))^1/n and then use the root test.

6. Don't forget about the limit comparison test: If you are stuck on a problem, try using the limit comparison test with a known convergent or divergent series. This can often simplify the problem and make it easier to determine convergence.

7. Practice, practice, practice: The more you practice solving convergence test problems, the more familiar you will become with the various techniques and tricks. So don't be afraid to try out different methods and keep practicing until you feel confident with these types of problems.

In conclusion, solving convergence test problems can be challenging, but with these tips and tricks, you should be able to approach them with more confidence and efficiency. Remember to always carefully read the problem, identify the type of series, and choose the most appropriate convergence test. And don't be afraid to use algebraic manipulation and comparison tests to help you solve the problem. Keep