• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Determine whether the series converges or diverges

  • Thread starter tnutty
  • Start date
327
1
1. Homework Statement

Determine whether the series converges or diverges.

[tex]\sum[/tex](1+5^n) / (1+6^n)

2. Homework Equations

limit comparison test, or just the comparison test.


3. The Attempt at a Solution

[attempt #1]

I have a couple of ways of trying to proves this.

1) (1+5^n) / (1+6^n) < (1 + 5^n) / 6^n =

(1/6)^n + (5/6)^n , which converges to 1/5 + 5 = 26/5

so since the series is bounded by 0 <= (1+5^n)/(1+6^n) <=26/5
it converges? This might not necessarily be true because a function might fluctuate between
those points until infinity, but I think that only cosine have this property. Thus this series converges.


[attempt # 2]

(1+5^n) / (1+6^n) =

1 / (1+6^n) + (5/6)^n

I know that (5/6)^n converges to 5, so

= 1/(1+6^n) + 5

now I apply the comparison test to 1/(1+6^n)

1/(1+6^n) < 1/(6^n), and this converges to 0.2

so ,

[0<= 1/(1+6^n) <= 0.2 ] + 5

so it converges since 1/(1+6^n) is bounded.


Either 1 right?
 

Dick

Science Advisor
Homework Helper
26,258
618
Re: Converge/diverge

Yes, use a comparison test. The first attempt is better. But if you are going to give values for the sum you should probably give a value for the lower limit on n. Is it 0 or 1 or 2 or what?
 

Related Threads for: Determine whether the series converges or diverges

Replies
15
Views
6K
Replies
3
Views
4K
Replies
2
Views
2K
Replies
3
Views
252
Replies
2
Views
502

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top