# Converges or diverges?

2020 Award
Homework Statement:
Sum n varying = [1,2,...,infinite(
3/(5^n) + 2/n
Relevant Equations:
3/(5^n) + 2/n
I know that it diverges, i dont know how to proof it:
We can decompose a sum in partial sums just if the two sums alone converges, so in this case we can not decompose in sum 3/5^n + sum 2/n, so how to proof that diverges just with the initial term?

scottdave
Homework Helper
Take each term. If you have (something converges) + (something diverges), what do you have?

How does the infinite sum of ## \frac {1}{n} ## behave?
Do you know how to prove that?

FactChecker
2020 Award
Take each term. If you have (something converges) + (something diverges), what do you have?

How does the infinite sum of ## \frac {1}{n} ## behave?
Do you know how to prove that?
There is a proof from a mathematician of the middle ages, like
1/1
1/1 + 1/2 = 1 + 1/2
1/1 + 1/2 + 1/3 + 1/4 > 1 + 1/2 + 1/4 + 1/4 > 1 + 1/2 + 1/2
...
I dont think that is a rigorous proof to the mathematic of our age, but is enough to the exercise.

About the first question, i think that no make sense the sum of a number with something that is not a number [like infinite], i would say that this sum just could diverge. I am sad because i dont know how to proof this, and since my intuition sometimes fool me, i am not certainly about my answer.

THere is some theorem that i can support my argument?

Mark44
Mentor
There is a proof from a mathematician of the middle ages, like
1/1
1/1 + 1/2 = 1 + 1/2
1/1 + 1/2 + 1/3 + 1/4 > 1 + 1/2 + 1/4 + 1/4 > 1 + 1/2 + 1/2
...
I dont think that is a rigorous proof to the mathematic of our age, but is enough to the exercise.
It would be rigorous if you could show that the sum of the first n terms > the sum of the first n terms of 1 + 1/2 + 1/2 + … .
LCSphysicist said:
About the first question, i think that no make sense the sum of a number with something that is not a number [like infinite]
The name of this symbol, ##\infty##, is infinity. The series ##\sum_{i = 1}^\infty \frac 1 n## is an example of an infinite series. Infinity is a noun, and infinite is an adjective that is used to describe a noun.
LCSphysicist said:
, i would say that this sum just could diverge. I am sad because i dont know how to proof this, and since my intuition sometimes fool me, i am not certainly about my answer.

THere is some theorem that i can support my argument?

LCSphysicist
2020 Award
It would be rigorous if you could show that the sum of the first n terms > the sum of the first n terms of 1 + 1/2 + 1/2 + … .
The name of this symbol, ##\infty##, is infinity. The series ##\sum_{i = 1}^\infty \frac 1 n## is an example of an infinite series. Infinity is a noun, and infinite is an adjective that is used to describe a noun.

I didnt really know, thank you for the corrections.

scottdave