- Evaluate
SUM(1/(n + n^2))from n = 1 to infty
this post was submitted on 08 May 2024
17 points (90.5% liked)
Daily Maths Challenges
189 readers
18 users here now
Share your cool maths problems.
Complete a challenge:
- Post your solution in comments, if it is exactly the same as OP's solution, let us know.
- Have fun.
Post a challenge:
- Doesn't have to be original, as long as it is not a duplicate.
- Challenges not riddles, if the post is longer than 3 paragraphs, reconsider yourself.
- Optionally include solution in comments, let it be clear this is not a homework help forums.
- Tag [unsolved] if you don't have a solution yet.
- Please include images, if your question includes complex symbols, attach a render of the maths.
Feel free to contribute to a series by DMing the OP, or start your own challenge series.
founded 6 months ago
MODERATORS
you are viewing a single comment's thread
view the rest of the comments
view the rest of the comments
solution
With partial fractions:1/(n + n²) = 1/(n(n+1)) = A/n + B/(n+1)
A(n+1) + Bn = 1
n = 0 gives A = 1, n = -1 gives B = -1
1/(n+n²) = 1/n - 1/(n+1)
Σ (n = 1 to ∞) 1/(n+n²) = Σ (n = 1 to ∞) 1/n - Σ (n = 1 to ∞) 1/(n+1)
= Σ (n = 1 to ∞) 1/n - Σ (n = 2 to ∞) 1/n
= 1/1 + Σ (n = 2 to ∞) 1/n - Σ (n = 2 to ∞) 1/n
= 1
Guessing this is the standard solution