Barrons AP Calculus

The series  therefore   converges.

(b) Since   the series  converges   by  the Alternating Series  Test,   the error   in

using   the first   n   terms   for the sum of  the whole   series  is  less    than    the

absolute    value   of  the (n  +   1)st    term.   Thus    the error   is  less    than    .

Solve   for n   using    :

The given   series  converges   very    slowly!

(c)  The     series  is  conditionally   convergent.     The     given

alternating  series  converges   since   the     nth     term    approaches  0   and    

. However, the nonnegative series diverges by the
Integral Test, since

(a) Solve   by  separation  of  variables:

Let c   =   e−^10 C;    then

Now use initial condition   y   =   2   at  t   =   0:

and the other   condition,  y   =   5   at  t   =   2,  gives

(b) Since   c   =   4   and     ln  2,  then    .