Question

Consider the series [infinity]Σn=1 1/n(n+1) Write the series using sigma notation. and

Write down the first four terms in the sequence of partial sums, and then find the exact formula for the N-th partial sum.
Use the exact formula to determine if the series converges or diverges. If it converges, find the value it converges to.

Answer 1

The first four terms are: 1/2, 2/3, 7/12 and 37/60.

The exact formula for the N-th partial sum is: SN = 1/2 - 1/(N+1).

The series converges to 1/2.

Step-by-step explanation:

The given series can be written in sigma notation as ∑ n=1 to ∞ 1/(n(n+1)).

The first four terms in the sequence of partial sums are:

S1 = 1/(1(1+1)) = 1/2

S2 = 1/(1(1+1)) + 1/(2(2+1))

= 1/2 + 1/6

= 2/3

S3 = 1/(1(1+1)) + 1/(2(2+1)) + 1/(3(3+1))

= 1/2 + 1/6 + 1/12

= 7/12

S4 = 1/(1(1+1)) + 1/(2(2+1)) + 1/(3(3+1)) + 1/(4(4+1))

= 1/2 + 1/6 + 1/12 + 1/20

= 37/60

The N-th partial sum can be found using the formula SN = 1/2 - 1/(N+1).

To determine if the series converges or diverges, we can take the limit of the N-th partial sum as N approaches infinity.

Taking the limit, we get limN->∞ SN = limN->∞ (1/2 - 1/(N+1)) = 1/2.

Therefore, the series converges to 1/2.