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What is the sum of the series \(\frac{1}{10^{2}}+\frac{1}{10^3}+\frac{2}{10^4}+\frac{3}{10^5}+\frac{5}{10^6}+\frac{8}{10^7}+\dots\)? (Note: The number is in Fibonacci Sequence)

 Apr 13, 2024
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This is a series where the numerator follows the Fibonacci sequence and the denominator is a geometric sequence with common ratio 1/10. To find the sum of such a series, we can use a technique involving manipulation and summation of geometric series.

 

Here's how to solve it:

 

Step 1: Splitting the Series

 

We can represent the series as the sum of two series:

 

S = (1/10^2 + 1/10^3 + 1/10^4 + ...) + (2/10^4 + 3/10^5 + 5/10^6 + ...)

 

Step 2: Recognizing Geometric Series

 

The first series is a geometric series with first term (a = 1/10^2) and common ratio (r = 1/10). The second series is another geometric series with first term (a = 2/10^4) and common ratio (r = 1/10).

 

Step 3: Find the Sum of Each Series

 

The formula for the sum (Sn) of a finite geometric series is:

 

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

 

where:

 

a is the first term

 

r is the common ratio

 

n is the number of terms

 

Step 4: Apply the Formula to Each Series

 

First Series:

 

S1 = (1/10^2) * (1 - (1/10)^n) / (1 - 1/10)

 

Second Series:

 

S2 = (2/10^4) * (1 - (1/10)^n) / (1 - 1/10)

 

Step 5: Note on Infinite Series

 

Since we're dealing with an infinite series (n tends to infinity), both (1/10)^n terms in the numerators approach zero. Therefore, we can simplify the expressions:

 

S1 ≈ (1/10^2) * (1 - 0) / (1 - 1/10) = 1/100

 

S2 ≈ (2/10^4) * (1 - 0) / (1 - 1/10) = 1/50

 

Step 6: Sum the Simplified Series

 

The total sum (S) of the original series is the sum of S1 and S2:

 

S = S1 + S2 ≈ 1/100 + 1/50 = 3/100

 

Answer:

 

Therefore, the sum of the series is 3/100.

 Apr 14, 2024

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