Question

$\lim _{\mathrm{n} \rightarrow \infty}\left\\{\frac{7}{10}+\frac{29}{10^{2}}+\frac{133}{10^{3}}+\ldots . .+\frac{5^{\mathrm{n}}+2^{\mathrm{n}}}{10^{\mathrm{n}}}\right\\}$ equals (A) \(3 / 4\) (B) 2 (C) \(5 / 4\) (D) \(1 / 2\)

Short answer

From the step-by-step solution, the limit of the given infinite series is equal to the sum of two separate geometric series. After calculating the sum of these two series, we find that the value of the limit is 3/4. So, the answer to this short answer question is: "3/4"

Step-by-step solution

Separate the terms into two geometric series

To do this, rewrite the given sequence as the sum of two separate series: one containing the terms \(\frac{5^n}{10^n}\) and the other containing the terms \(\frac{2^n}{10^n}\). So, the given series becomes: \(\lim_{n \rightarrow \infty} \left( \sum_{n=1}^{\infty} \left( \frac{5^n}{10^n} + \frac{2^n}{10^n}\right)\right)\).

Calculate the sum of the first geometric series

The first geometric series is \(\sum_{n=1}^{\infty} \frac{5^n}{10^n}\). To find its sum, find the first term (\(a_1\)) and the common ratio (\(r_1\)). For this series, \(a_1 = \frac{5^1}{10^1} = \frac{1}{2}\) and \(r_1 = \frac{5}{10} = \frac{1}{2}\). Now, use the formula for the sum of an infinite geometric series: \(S_1 = \frac{a_1}{1 - r_1}\). Substitute the values we've found: \(S_1 = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{1}{2}\).

Calculate the sum of the second geometric series

The second geometric series is \(\sum_{n=1}^{\infty} \frac{2^n}{10^n}\). Again, find the first term (\(a_2\)) and the common ratio (\(r_2\)). For this series, \(a_2 = \frac{2^1}{10^1} = \frac{1}{5}\) and \(r_2 = \frac{2}{10} = \frac{1}{5}\). Now, use the formula for the sum of an infinite geometric series: \(S_2 = \frac{a_2}{1 - r_2}\). Substitute the values we've found: \(S_2 = \frac{\frac{1}{5}}{1 - \frac{1}{5}} = \frac{1}{4}\).

Add the sums of the two geometric series

Now that we have the sums of the two geometric series, we can add them together to find the value of the given limit. \(\lim_{n \rightarrow \infty} \left(\sum_{n=1}^{\infty} \left( \frac{5^n}{10^n} + \frac{2^n}{10^n} \right)\right) = S_1 + S_2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}\). So the correct answer is (A) \(3 / 4\).