To determine if the given series [tex]\(\frac{1}{1.3}+\frac{1}{2.5}+\frac{1}{3.7}+\frac{1}{4.9}+\ldots\)[/tex] equals [tex]\(2(1 - \ln 2)\)[/tex], let's examine the terms of the series and whether their sum matches the expression on the right-hand side.
### Step-by-Step Solution
1. Collection of Terms:
The series begins with the following terms:
[tex]\[
\frac{1}{1.3}, \frac{1}{2.5}, \frac{1}{3.7}, \frac{1}{4.9}
\][/tex]
2. Calculating Each Term:
- First term:
[tex]\[
\frac{1}{1.3} \approx 0.7692307692307692
\][/tex]
- Second term:
[tex]\[
\frac{1}{2.5} = 0.4
\][/tex]
- Third term:
[tex]\[
\frac{1}{3.7} \approx 0.27027027027027023
\][/tex]
- Fourth term:
[tex]\[
\frac{1}{4.9} \approx 0.2040816326530612
\][/tex]
3. Summing the Terms:
To find the sum of the series up to the given terms:
[tex]\[
0.7692307692307692 + 0.4 + 0.27027027027027023 + 0.2040816326530612 \approx 1.6435826721541007
\][/tex]
4. Comparing with the Given Expression:
The given expression is [tex]\(2(1 - \ln 2)\)[/tex]. To see if our calculated sum is equal to it, let's first compute the numerical value of the right-hand side.
- Calculating [tex]\(1 - \ln 2\)[/tex]:
[tex]\[
\ln 2 \approx 0.6931471805599453
\][/tex]
[tex]\[
1 - 0.6931471805599453 \approx 0.3068528194400547
\][/tex]
- Then multiply by 2:
[tex]\[
2 \times 0.3068528194400547 \approx 0.6137056388801094
\][/tex]
Since the sum of the series up to the given terms is approximately [tex]\(1.6435826721541007\)[/tex] and the evaluated right-hand side is about [tex]\(0.6137056388801094\)[/tex], they are not equal.
### Conclusion
The sum of the given terms of the series [tex]\(\frac{1}{1.3}+\frac{1}{2.5}+\frac{1}{3.7}+\frac{1}{4.9}+\ldots\)[/tex] does not equal [tex]\(2(1 - \ln 2)\)[/tex]. The value of the sum of the series terms is approximately [tex]\(1.6435826721541007\)[/tex], while the expression evaluates to approximately [tex]\(0.6137056388801094\)[/tex]. Hence, the statement is false.
