Proof: Divergence of 3/5^n + 2/n Sum

[LCSphysicist]

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 don't 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]

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?

 

[LCSphysicist]

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 don't 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 don't 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?

 

[scottdave]

Perhaps something like this is what you're looking for. https://math.berkeley.edu/~kmill/math1bfa15/11.2.note.pdf

 

[Mark44]

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 don't 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 + … .

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.

, i would say that this sum just could diverge. I am sad because i don't 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]

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]

Do you know Calculus? If so, you could use the Integral Test to test $\frac {1}{n}$

 

[scottdave]

Pauls Online notes has some great explanation of Calculus concepts, including the Integral Test. http://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx