Solution Found!
Telescoping series For the following
Chapter 11, Problem 55E(choose chapter or problem)
47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges.
\(\sum_{k=1}^{\infty}\left(\frac{1}{\sqrt{k+1}}-\frac{1}{\sqrt{k+3}}\right)\)
Questions & Answers
QUESTION:
47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges.
\(\sum_{k=1}^{\infty}\left(\frac{1}{\sqrt{k+1}}-\frac{1}{\sqrt{k+3}}\right)\)
ANSWER:Problem 55E
Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or stale that the series diverges.
Answer ;
Step 1 ;
The given Telescoping series is -
)
In this problem we have to find the formula for term
in
-
)
and then we have to evaluate or we have state that the series diverges.
Consider -
) ……….(1)
Let us first find the term of the sequence of partial sums
=
-
) ………….(2)
Substitute values for we get
=
-
+
-
+
-
………+
-
+
-
.
Cancelling the like terms with opposite sign we get,
=
=
+
-(
+
).
Thus the term in the series is
=
+
-(
+
).