- #1
Platformance
- 33
- 0
Homework Statement
Show that:
[itex]\sum \frac{3}{n^{2} + 1}[/itex]
converges from n = 1 to ∞
Homework Equations
If Ʃbn converges, and Ʃan < Ʃbn.
Ʃan also converges.
The Attempt at a Solution
[itex]\sum \frac{1}{n^{2}}[/itex] converges
[itex]\sum \frac{3}{n^{2} + 1}[/itex] = 3 * [itex]\sum \frac{1}{n^{2} + 1}[/itex]
[itex]\sum \frac{1}{n^{2} + 1}[/itex] < [itex]\sum \frac{1}{n^{2}}[/itex] for all n from 1 to ∞.
Therefore [itex]\sum \frac{1}{n^{2} + 1}[/itex] converges.
Therefore [itex]\sum \frac{3}{n^{2} + 1}[/itex] also converges.
The problem I am having is if the 3 remained in the summation.
[itex]\sum \frac{3}{n^{2} + 1}[/itex] is not less than [itex]\sum \frac{1}{n^{2}}[/itex] for all n from 1 to ∞.
Why does placing the 3 outside the summation make the problem work?