series ∑ (10⋅20⋅30...(10k))/(10k)! converges or diverges.
series ∑ (10⋅20⋅30...(10k))/(10k)! converges or diverges.
Answer
Explanation:
Step1: Define the series term
Let $a_k=\frac{10\cdot20\cdot30\cdots(10k)}{(10k)!}$.
Step2: Apply the ratio - test
Calculate $\lim_{k\rightarrow\infty}\left|\frac{a_{k + 1}}{a_k}\right|$. First, find $a_{k+1}=\frac{10\cdot20\cdot30\cdots(10k)\cdot(10(k + 1))}{(10(k + 1))!}=\frac{10\cdot20\cdot30\cdots(10k)\cdot(10k+10)}{(10k + 10)!}$. Then $\frac{a_{k + 1}}{a_k}=\frac{\frac{10\cdot20\cdot30\cdots(10k)\cdot(10k + 10)}{(10k+10)!}}{\frac{10\cdot20\cdot30\cdots(10k)}{(10k)!}}$. Simplify the ratio: [ \begin{align*} \frac{a_{k + 1}}{a_k}&=\frac{10\cdot20\cdot30\cdots(10k)\cdot(10k + 10)\cdot(10k)!}{10\cdot20\cdot30\cdots(10k)\cdot(10k+10)!}\ &=\frac{10k + 10}{(10k+1)(10k + 2)\cdots(10k+10)} \end{align*} ]
Step3: Find the limit
As $k\rightarrow\infty$, the degree of the numerator is 1 and the degree of the denominator is 10. So $\lim_{k\rightarrow\infty}\left|\frac{a_{k + 1}}{a_k}\right| = 0$. Since $\lim_{k\rightarrow\infty}\left|\frac{a_{k + 1}}{a_k}\right|=0<1$, by the ratio - test, the series $\sum a_k$ converges.
Answer:
The series converges.