Prove limit with convergence tests

In summary, the conversation discusses using convergence tests to prove the limit of two sequences. The first sequence is limn→∞ n*q^n=0,|q|<1 and the second is limn→∞ 2*n/n!. The speaker was able to use the "ratio test" to show that the second sequence converges to 0, but is unsure how to prove the limit of the first sequence. They also mention using the squeeze theorem to solve the second sequence.
  • #1
esuahcdss12
10
0
I need to prove that the limit of the sequence is as shown(0):

1.limn→∞ n*q^n=0,|q|<1
2.limn→∞ 2*n/n!
but I need to do this using the convergence tests. With the second sequence I tried the "ratio test", and I got the result

limn→∞ 2/n+1
which means that L in the ratio test is 0 and so it proves that the sequence converges, but how now should i prove that the limit is indeed 0? I can't use the L'Hopital's rule.

and for the first sequence I am not sure where to start.

can you help please?
 
Last edited:
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  • #2
It looks like question 2 has a typo.
 
  • #3
greg1313 said:
It looks like question 2 has a typo.

Fixed.
can anyone help with the first sequence ?
the second I manged to solve with the squeeze theory
 

What is the purpose of proving a limit with convergence tests?

The purpose of proving a limit with convergence tests is to determine whether a given sequence or series converges or diverges. This is important in mathematics and science because it helps us understand the behavior of functions and make predictions about their values.

What is the difference between a sequence and a series?

A sequence is a list of numbers in a specific order, while a series is the sum of all the terms in a sequence.

What are some common convergence tests used to prove limits?

Some common convergence tests include the comparison test, the ratio test, and the root test. The limit comparison test and the integral test are also frequently used.

How do these convergence tests work?

The comparison test compares a given series to a known convergent or divergent series to determine its convergence. The ratio and root tests use the ratio and root of the terms in a series to determine convergence. The limit comparison test compares the limit of a given series to the limit of a known convergent series. The integral test uses the integral of a function to determine the convergence of the series.

What are some applications of proving limits with convergence tests?

Proving limits with convergence tests has various applications in fields such as physics, engineering, and economics. For example, in physics, convergence tests can be used to determine the behavior of electric and magnetic fields. In engineering, they can be used to analyze the stability of structures. In economics, they can be used to predict the growth of investments over time.

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