Sammy asked in Science & MathematicsMathematics · 7 months ago

Limiting Sum maths in focus question, find t?

Hi can anyone help, I am really not too sure how they got this answer, the question is

For what values of k does the limiting sum exist for the series …? k+ k^2 + k^3....

Update:

K not T!!!

Relevance
• 7 months ago

S = k + k^2 + k^3 + k^4 + ...

S = k + k * (k + k^2 + k^3 + .....

S = k + k * S

S - S * k = k

S * (1 - k) = k

S = k / (1 - k)

k / (1 - k) = k + k^2 + k^3 + ....

k / (1 - k)

For infinite geometric sums, when the common ratio is between -1 and 1, the sum converges. In our sum, k is the common ratio between any 2 terms. Therefore, when -1 < k < 1, then we have a converging sum, and it converges to k / (1 - k)

• Mewtwo
Lv 5
7 months agoReport

Very well explained.

• 7 months ago

The series is:

k + k² + k³ + k⁴ + ...

This can be rewritten as:

k(k)⁰ + k(k)¹ + k(k)² + k(k)³ + ...

This is a geometric series, where the first term is k and the common ratio is k.

The sum of the first n terms of a geometric series is:

S = a (1 - r^n) / (1 - r)

In this case, a = k and r = k:

S = k (1 - k^n) / (1 - k)

The limit as n approaches infinity:

lim(n→∞) S =

lim(n→∞) k (1 - k^n) / (1 - k) =

k / (1 - k) lim(n→∞) (1 - k^n)

If k is a fraction, then k^n will approach 0 as n approaches infinity, so the limit will exist. Otherwise, k^n will grow to infinity, and the limit will not exist.
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