+302 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)
+1911 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.
