How do you use the ratio test to test the convergence of the series #∑(5^k+k)/(k!+3)# from n=1 to infinity?

Answer 1

See below

Trying to add some understanding about that series. #∑(5^k+k)/(k!+3) = sum 5^k/(k!+3) + sum 1/((k-1)!+3/k)< sum 5^k/(k!) + sum 1/((k-1)!) = e^5+e^1# so the series converges for a number #gamma# such that
#gamma < e^5+e^1#
Sign up to view the whole answer

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

Sign up with email
Answer 2

To use the ratio test to test the convergence of the series ( \sum_{n=1}^{\infty} \frac{5^k + k}{k! + 3} ):

  1. Compute the limit of the absolute value of the ratio of consecutive terms: [ \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| ]

  2. If the limit is less than 1, the series converges. If it's greater than 1 or the limit does not exist, the series diverges. If the limit equals 1, the test is inconclusive.

Let's apply this to the given series:

[ a_k = \frac{5^k + k}{k! + 3} ]

[ \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{\frac{5^{k+1} + (k+1)}{(k+1)! + 3}}{\frac{5^k + k}{k! + 3}} \right| ]

[ = \lim_{k \to \infty} \left| \frac{(5^{k+1} + (k+1))(k! + 3)}{(5^k + k)((k+1)! + 3)} \right| ]

[ = \lim_{k \to \infty} \left| \frac{5^{k+1}k! + 5^{k+1} + k(k! + 3) + (k+1)(k! + 3)}{5^kk! + 5k! + k(k+1)! + 3(k+1)! + 3} \right| ]

[ = \lim_{k \to \infty} \left| \frac{5^{k+1}k! + 5^{k+1} + k! + 3k + k + 1}{5^kk! + 5k! + k! + 3k! + 3} \right| ]

[ = \lim_{k \to \infty} \left| \frac{(5^{k+1} + 1)k! + (k + 3)}{(5^k + 1)k! + (3k! + 3)} \right| ]

[ = \lim_{k \to \infty} \left| \frac{5^{k+1} + 1}{5^k + 1} \right| = 5 ]

Since the limit is greater than 1, the series diverges by the Ratio Test.

Sign up to view the whole answer

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

Sign up with email
Answer from HIX Tutor

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7