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Sagot :
There is no value of c such that the infinite series converges.
How to analyze the converge of an infinite series
Convergence is a characteristic of infinite series, in which a sum tends to be a given number when n tends to +∞.
In this question we must prove for which values of c the infinite series may converge.
There are different criteria to prove whether a given series converges or not, one of the most used criteria is the ratio criterion, which states that:
[tex]r = \frac{a_{k+1}}{a_{k}}[/tex], where the series converges if and only if r > 1. (1)
If we know that [tex]a_{k} = k\cdot \ln c[/tex], then the rational formula is:
[tex]r = \frac{(k+1)\cdot \ln c}{k\cdot \ln c}[/tex]
[tex]r = 1 + \frac{1}{k}[/tex]
We notice that resulting expression does not depend on c and is greater than 1. Therefore, there is no value of c such that the infinite series converges. [tex]\blacksquare[/tex]
Remark
If [tex]c[/tex] is a positive real number and [tex]a_{k} = k\cdot \ln c[/tex], for what values of [tex]c[/tex], if any, does the infinite series [tex]\sum_{k=1}^{\infty} a_{k}[/tex] converge?
To learn more on series, we kindly invite to check this verified question: https://brainly.com/question/10813422
There is no values of c, for the infinite series ∑k=1[infinity]ak to be converge when c is a positive real number.
What do you mean by infinite series?
The infinite series is the sum of addition of a sequence in which the number of terms present are infinite.
[tex]a_1+a_2+a_3....+a_n[/tex]
Here, ([tex]a_n[/tex]) is the nth term of the sequence. For infinite series, the ratio criterion can be given as,
[tex]r=\dfrac{a_{k+1}}{a_k}[/tex]
If c is a positive real number and
[tex]a_k=k\ln c[/tex]
Put the value in above shown ratio,
[tex]r=\dfrac{(k+1)\ln c}{k\ln c}\\r=\dfrac{k\ln c}{k\ln c}+\dfrac{\ln c}{k\ln c}\\r=1+\dfrac{1}{k}[/tex]
With the value of r, we can say whether the series is converge or not. This value of r does not depends on c.
Thus, there is no values of c, for the infinite series ∑k=1[infinity]ak to be converge when c is a positive real number.
Learn more about the infinite series here;
https://brainly.com/question/26133507
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