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Chapter 1: 5P (page 17)

In the following problems, find the limit of the given sequence asn鈫�鈭�

10nn!

Short Answer

Expert verified

The limit of the expression 10nn! is 0

Step by step solution

01

Definition of limits

A limit of a function/expression/sequence is used when the value of the function/sequence cannot be calculated directly or is difficult to calculate at a particular value. It is generally defined as the output obtained when the input is very close to the input value.

01

Given parameters

The given expression is 10nn!and the limit is n鈫�鈭�

01

Ratio Test

Ratio test is a test conducted to find whether any series is converging or diverging.

01

Solve the limits

Let

Apply the ratio test in the given expression, that is

Solve the obtained limit.

It can be observed that the value ofis 0 which is less than 1, which suggest that the given series is converging and thus the value of limit


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Most popular questions from this chapter

鈭�n=1鈭�鈥�x2+1n2n3n+n3

f(x)=ex,a=3.

Prove that an absolutely convergent series 鈭�m=1鈭�anis convergent. Hint: Putbn=an+|an|. Then theare nonnegative; we have|bn|鈮�2|an|andan=bn-|an|

(a) Using computer or tables (or see Chapter 7,Section 11),verify that鈭�n=1鈭�(1/n2)=蟺26=1.6449=,and also verify that the error in approximating the sum of the series by the first five terms is approximately 0.1813.

(b) By computer or tables verify that 鈭�n=1鈭�(1/n2)(1/2)n=蟺212-(1/2)(ln2)2=0.5815+

the sum of the first five terms is0.5815+

(c) Prove theorem (14.4). Hint: The error is |鈭�N+1鈭�anxn|.

Use the fact that the absolute value of a sum is less than or equal to the sum of the absolute values. Then use the fact that |an+1|鈮�|an|to replace all anby aN+1 , and write the appropriate inequality. Sum the geometric series to get the result.

By computer or tables, find the exact sum of each of the following series.

(a)鈭�n=1鈭�n(n2+1)2(b)鈭�n=2鈭�Innn2(c)鈭�n=1鈭�1nn
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