Consider the infinite series sigmainfinityn1 3n 3n 1 Write

Consider the infinite series sigma^infinity_n=1 (3/n - 3/n + 1). Write the first 3 partial sums, S_1, S_2, S_3. [NOTE! remember that S_1 = a_1, S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3.] Find the limit lim_n rightarrow infinity S_n. This will be the sum of the series.

Solution

1) S₁ =(3_/1 -3_/2) =3_/2

S₂=(3_/1 -3_/2) +(3_/2 -3_/3) =2

S₃=(3_/1 -3_/2) +(3_/2 -3_/3)+(3_/3 -3_/4) =9_/4

b)S_n=(3_/1 -3_/2) +(3_/2 -3_/3)+(3_/3 -3_/4)+..........+(3_/n -3_/n+1) =3-(3_/n+1)

S_n=(3n_/n+1)

lim_n->S_n

=lim_n->(3n_/n+1)

=lim_n->(3_/1+(1/n))

=(3_/1+(0))

=3