Yahoo Answers: Answers and Comments for Limiting Sum maths in focus question, find t? [Mathematics]
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From Sammy
enUS
Sun, 16 Jun 2019 01:22:51 +0000
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Yahoo Answers: Answers and Comments for Limiting Sum maths in focus question, find t? [Mathematics]
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From Captain Matticus, LandPiratesInc: S = k + k^2 + k^3 + k^4 + ...
S = k + k * (k +...
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https://answers.yahoo.com/question/index?qid=20190616012251AA2Z6Qq
Sun, 16 Jun 2019 02:32:00 +0000
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)

From Some Body: The series is:
k + k² + k³ + k⁴ + ...
This ca...
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https://answers.yahoo.com/question/index?qid=20190616012251AA2Z6Qq
Sun, 16 Jun 2019 01:52:24 +0000
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.

From D g: you can see that if the k = 1 or larger the...
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Sun, 16 Jun 2019 01:28:25 +0000
you can see that if the k = 1 or larger then each successive value gets larger so it will never have a limit
if the k is less than 1 then each successive value gets smaller and so if something adds to something smaller than before there is eventually a limit to the sum
since the a fraction of one times itself is always smaller this works for positive or negative