Beeebooo A.

asked • 07/08/15# Calculate S3,S4, and S5 and then find the sum for the telescoping series S=?n=68(1n+1-1n+2) S3 = S4 = S5 = S =

S=∑(1/n+1−1/n+2),n=0..infinity

S3 =

S4 =

S5 =

S =

## 1 Expert Answer

**OK. Here is my final attempt:**

**Σ[1/(n+1)-1/(n+2)] from n = 6 to ∞**

**= [1/7-1/8] + [1/8-1/9] + [1/9 - 1/10] + ...**

**sum of first n terms**

**= [1/7-1/8]+[1/8-1/9]+[1/9-1/10]+...+ [1/(n+6) - 1/(n+7)]**

**= 1/7 - 1/(n+7)**

**S**

_{3}= sum of first 3 terms = 1/7 - 1/10 = 3/70**S**

_{4}= sum of first 4 terms = 1/7 - 1/11 = 4/77**S**

_{5}= sum of first 5 terms = 1/7 - 1/12 = 5/84**The sum, S, of the entire series is given by:**

**S = lim (S**

_{n}) as n →∞ = lim[1/7 - 1/(n+7)] as n→∞**= 1/7 - 0 = 1/7**

_{ }Beeebooo A.

07/08/15

Beeebooo A.

07/08/15

Beeebooo A.

07/08/15

Beeebooo A.

07/08/15

Mark M.

**OK. Fire away.**

07/08/15

Beeebooo A.

(a) Find a function f(k) such that

2−4+6−8+10=∑f(k), k=1 to 5

f(k) =

(b) Find a function g(k) such that

2−4+6−8+10=∑g(k), k=0 to 4

g(k) =

07/08/15

Mark M.

**2, 4, 6, 8, ... are multiples of 2**

**Use (-1)**

^{k}or (-1)^{k}^{+1}to make the signs alternate**(a). Let f(k) = (-1)**

^{k+1}(2k)**Then Σ[(-1)**

^{k+1}(2k)] from k = 1 to k = 5**= (-1)**

^{2}(2)+(-1)^{3}(4)+(-1)^{4}(6)+(-1)^{5}(8)+(-1)^{6}(10)**= 2-4+6-8+10**

**(b). Let g(k) = (-1)**

^{k}(2k)**Σ[(-1)**

^{k}(2(k+1)) from k = 0 to k = 4**= (-1)**

^{0}(2) + (-1)^{1}(4) + (-1)^{2}(6) + (-1)^{3}(8) + (-1)^{4}(10)**= 2-4+6-8+10**

07/08/15

Beeebooo A.

07/08/15

Mark M.

**Correction: g(k) = (-1)**

^{k}(2(k+1)), not (-1)^{k}(2k).07/08/15

Jacob L.

= [1/7-1/8]+[1/8-1/9]+[1/9-1/10]+...+ [1/(n+6) - 1/(n+7)]

= 1/7 - 1/(n+7)"

09/29/15

Mark M.

09/29/15

Jacob L.

09/29/15

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Stephanie M.

do not have to answerthese questions, and they don't get paid to do so. They do so voluntarily, on their own time, in order to help students in need. If you don't check the correctness of your question, you'll get incorrect answers. If you don't format your question in an understandable way, you'll get no answers. If you're rude or unhelpful to tutors, they likely won't try to answer your questions in the future.07/08/15