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Chapter 9: Problem 56

Sequence and Series Describe the difference between \(\lim _{n \rightarrow \infty} a_{n}=5\) and \(\sum_{n=1}^{\infty} a_{n}=5\)

Short Answer

Expert verified
The limit of a sequence refers to the value the terms of the sequence are approaching as \(n\) reaches infinity, while the sum of a series is the total value obtained by adding up all the terms of the series. In this case, \(\lim _{n \rightarrow \infty} a_{n}=5\) means the sequence \(a_n\) approaches 5 as \(n\) grows larger, while \(\sum_{n=1}^{\infty} a_{n}=5\) means the total sum of all terms of the series \(a_n\) is equal to 5.

Step by step solution

01

Understanding Limit of a Sequence

The limit of a sequence refers to the value that the terms of a sequence approach as the index goes indefinitely to infinity. In this case, \(\lim _{n \rightarrow \infty} a_{n}=5\) denotes that the sequence \(a_n\) is approaching the value 5 as \(n\) approaches infinity.
02

Understanding Sum of a Series

The sum of a series is the result obtained by adding up all the terms of a series. The notation \(\sum_{n=1}^{\infty} a_{n}=5\) means that when we add all the terms of the series \(a_n\) from \(n=1\) to infinity, the summation is equal to 5.
03

Differences between limit of a sequence and Sum of a series

1. The limit of a sequence is a single value that the terms of a sequence approach as the index goes to infinity, while the sum of a series is the total computed by adding up those particular terms of a sequence. 2. The limit of a sequence does not involve the operation of addition, while a series inherently involves addition of sequence terms. 3. The limit of a sequence applies to the terms of a sequence individually while the summation applies to the total of all sequence terms.

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