(b) Discuss the convergence of the series 09 in its dependence on q. Determine its sum for the values of q such that the series converges. (c) Geometric series can be used to obtain expressions for rational numbers, given in decimal form, as the ratio of two coprime integers. For x = 0.363636 = 0.36 we can also write x = 36/100 +36/10000+36/1000000+... Use this to express x as the ratio of two coprime integers.
(b) Discuss the convergence of the series 09 in its dependence on q. Determine its sum for the values of q such that the series converges. (c) Geometric series can be used to obtain expressions for rational numbers, given in decimal form, as the ratio of two coprime integers. For x = 0.363636 = 0.36 we can also write x = 36/100 +36/10000+36/1000000+... Use this to express x as the ratio of two coprime integers.
Chapter8: Sequences, Series,and Probability
Section8.1: Sequences And Series
Problem 9ECP: For the series i=1510i find (a) the fourth partial sum and (b) the sum. Notice in Example 9(b) that...
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can you please do part b and c , written out please not typed and provide expanations
Transcribed Image Text:B2. (a) Consider the infinite series
where q E R.
=
Show - using induction or otherwise - that for the partial sums sn
we have
Σat,
k=0
Sn
...=
1- qn+1
9
(b) Discuss the convergence of the series
its sum for the values of q such that the series converges.
in its dependence on q. Determine
(c) Geometric series can be used to obtain expressions for rational numbers, given in
decimal form, as the ratio of two coprime integers.
For x = 0.363636. 0.36 we can also write
Σ=o q*,
k=0
x = = 36/100 + 36/10000+36/1000000+.
Use this to express x as the ratio of two coprime integers.
(d) For x ≥ 0, a function f is defined as an infinite series
∞
f(x) = Σ (2v)*.
k=0
Using the ratio test or otherwise, show that the series converges for 0 < x < 1/4.
What can you say about the convergence at x = 1/4?
(e) Determine f(x) (i.e., the value of the sum) for 0 < x < 1/4.
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