# How do you use the ratio test to test the convergence of the series #∑3^k/((k+1)!)# from n=1 to infinity?

The series converges absolutely.

The ratio test states the following:

THEN, if the limit exists:

Before dividing, it is useful to consider that:

Now we can divide:

#a_{k+1}/a_k= {3*3^k}/{(k+2)(k+1)!} * {(k+1)!}/3^k

We can simplify a lot of stuff:

{3*color(red)(cancel(3^k))}/{(k+2)color(blue)(cancel((k+1))!)} * {color(blue)cancel((k+1)!)}/color(red)(cancel(3^k))

Now we can easily take the limit:

By signing up, you agree to our Terms of Service and Privacy Policy

To use the ratio test to test the convergence of the series ( \sum \frac{3^k}{(k+1)!} ) from ( n = 1 ) to ( \infty ), you calculate the limit:

[ L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| ]

where ( a_k = \frac{3^k}{(k+1)!} ).

Substitute ( a_{k+1} ) and ( a_k ) into the limit expression and simplify:

[ L = \lim_{k \to \infty} \left| \frac{\frac{3^{k+1}}{(k+2)!}}{\frac{3^k}{(k+1)!}} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{3^{k+1}(k+1)!}{(k+2)! \cdot 3^k} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{3 \cdot (k+1)!}{(k+2)(k+1)!} \right| ]

[ L = \lim_{k \to \infty} \left| \frac{3}{k+2} \right| ]

[ L = 0 ]

Since the limit ( L ) is less than 1, by the ratio test, the series ( \sum \frac{3^k}{(k+1)!} ) converges absolutely.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do I find a formula for #s_n# for the sequence -2, 1, 6, 13, 22,...?
- How do you test the series #Sigma lnn/n# from n is #[1,oo)# for convergence?
- How do you test the alternating series #Sigma (-1)^nsqrtn# from n is #[1,oo)# for convergence?
- Determine the values of #p# for which the integral below is convergent?
- How do you use the integral test to determine the convergence or divergence of #1+1/(2sqrt2)+1/(3sqrt3)+1/(4sqrt4)+1/(5sqrt5)+...#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7