Using the integral test, how do you show whether #sum (n + 2) / (n + 1)# diverges or converges from n=1 to infinity?

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Answer 1 · 2015-07-04T19:52:18

The series diverges.

Without the integral test:

The terms do not go to #0# as #nrarroo# (they go to #1#), so the series diverges.

If we must use the integral test let's do so as quickly as possible.

#f(x) = (x+2)/(x+1) = (x+1+1)/(x+1) = 1+1/(x+1)#

#f'(x) = -1/(x+1)^2# which is negative, so #f# is a decreasing function.

#int_1^oo f(x) dx = lim_(brarroo) int_1^b (1+1/(x+1))dx#

# = lim_(brarroo) (x+ln abs(x+1))]_1^b#

# = lim_(brarroo)[ (b+ln(b+1)) - (1+ln2)]#

# = oo + oo -1-ln2 = oo#

The series diverges.