0. 1/3 points details my notes scalcet9 11.6.ae.004. example video example test the convergence of the…

0. 1/3 points details my notes scalcet9 11.6.ae.004. example video example test the convergence of the series sum(n = 1 to infinity) ((7n + 3)/(12n + 3))^n. solution a_n = ((7n + 3)/(12n + 3))^n nth - root(|a_n|) = 7+(3/n)/12+(3/n) -> < 1 thus the given series is convergent by the root test. need help? read it submit answer

0. 1/3 points details my notes scalcet9 11.6.ae.004. example video example test the convergence of the series sum(n = 1 to infinity) ((7n + 3)/(12n + 3))^n. solution a_n = ((7n + 3)/(12n + 3))^n nth - root(|a_n|) = 7+(3/n)/12+(3/n) -> < 1 thus the given series is convergent by the root test. need help? read it submit answer

Answer

Explanation:

Step1: Recall the root - test formula

For a series $\sum_{n = 1}^{\infty}a_n$, we consider $\lim_{n\rightarrow\infty}\sqrt[n]{\vert a_n\vert}$. Here $a_n=\left(\frac{7n + 3}{12n+3}\right)^n$, so $\sqrt[n]{\vert a_n\vert}=\frac{7n + 3}{12n+3}$.

Step2: Simplify the expression

Divide both the numerator and denominator of $\frac{7n + 3}{12n+3}$ by $n$: $\frac{7n+3}{12n + 3}=\frac{7+\frac{3}{n}}{12+\frac{3}{n}}$.

Step3: Find the limit as $n\rightarrow\infty$

As $n\rightarrow\infty$, $\frac{3}{n}\rightarrow0$. So $\lim_{n\rightarrow\infty}\frac{7+\frac{3}{n}}{12+\frac{3}{n}}=\frac{7 + 0}{12+0}=\frac{7}{12}<1$.

Answer:

The series $\sum_{n = 1}^{\infty}\left(\frac{7n + 3}{12n+3}\right)^n$ is convergent.