Is the function (1+e^(x^2))/(x+1) convergent or divergent? Please explain!?
2016-03-29 10:19 pm
The function is (1+e^(x^2))/(x+1) . The lower limit is 1 and upper limit is infinity. Any clue on how to even start this problem???
回答 (1)
2016-03-29 11:21 pm
Note that for all x > 1:
(1 + e^(x^2))/(x + 1)
> (0 + e^(x^2))/(x + x)
= e^(x^2)/(2x)
> e^(x^2)/(2xe^(x^2))
= 1/(2x).
Since ∫(x = 1 to ∞) dx/(2x) = (1/2) ln |x| {for x = 1 to ∞} = ∞ (divergent), we conclude that ∫(x = 1 to ∞) (1 + e^(x^2)) dx/(x + 1) is divergent by the Comparison Test.
I hope this helps!
(1 + e^(x^2))/(x + 1)
> (0 + e^(x^2))/(x + x)
= e^(x^2)/(2x)
> e^(x^2)/(2xe^(x^2))
= 1/(2x).
Since ∫(x = 1 to ∞) dx/(2x) = (1/2) ln |x| {for x = 1 to ∞} = ∞ (divergent), we conclude that ∫(x = 1 to ∞) (1 + e^(x^2)) dx/(x + 1) is divergent by the Comparison Test.
I hope this helps!
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