Question
- Tiven:
- Determie the value(s) of
for uhich the series will converge (4)
12 Calculate the value of
- Given the convergent geometrie seves:
;
- 1 Hebermine the possible vaine(s) of
(3) 22 If
culate
12 Calculate the value of
Ask by Gibson Marshall. in South Africa
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Problem 1:
- The series converges when
. - When
, .
Problem 2:
- The series converges when
. - When
, or approximately 65.33.
Solution

Sign in to Unlock Answers for Free!
A Learning Platform Trusted by Millions of Real Students and Teachers.

Answered by UpStudy AI and reviewed by a Professional Tutor
Like
Beyond the Answer
To determine the values of
for which the series converges, we can utilize the concept of a power series. The series you have resembles a polynomial in terms of
. Generally, a power series converges for values of
within its radius of convergence. For a series of the form
, convergence typically requires that the
th term approaches
as
approaches infinity. In this case, you might find convergence for
.
Now, if
, this can be solved using the formula for the sum of an infinite series. The sum of an infinite geometric series is given by
, where
is the first term and
is the common ratio. You’d set this up based on your series, solving
to find the required
.
Now, for the convergent geometric series you’ve provided
, the common ratio
can be expressed in terms of
. Specifically, the ratio between the second term and the first term gives
, and between the third and the second gives
as well. Setting
provides the crucial inequality
.
Lastly, if
, you can substitute it into the formula for the sum of the geometric series
. Plug in the values, and you’ll have
to determine the series sum under this condition.