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Chapter 12: Problem 24

Determine whether the series converges or diverges. $$\sum \frac{11}{1+100^{-2}}$$

Short Answer

Expert verified
The series diverges as it is a non-zero constant series.

Step by step solution

01

Understanding the Sequence

We are given the sequence \(\sum \frac{11}{1+100^{-2}}\). First, we need to understand that this sequence is actually a constant. That's because it's not dependent on the variable that usually sums up the sequence, for example the integer 'n'.
02

Identify Type of Sequence

Since there is no variable in the sequence, we conclude that it is a constant sequence. For each term in the sequence, the value is the same, it does not depend on the term number neither it progresses towards a certain number.
03

Determine Convergence or Divergence

A constant series diverges, unless the constant is zero. The exception is due to the fact that a zero sequence, which only contains zeros, has a finite sum. In this case, the constant is not zero, therefore, the series diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divergence
When we discuss whether a series converges or diverges, we're talking about what happens when we add an infinite number of terms together. A series diverges if these added terms do not approach a specific finite value. In simpler terms, think about a diverging series as a recipe that keeps adding ingredients forever, never reaching a specific taste or end point. This series keeps growing larger or doesn't settle around any number.

In mathematical terms, if the sum of the terms of the series isn't heading towards any particular value, it is diverging. If you imagine counting forever without stopping, and never reaching a limit, that's divergence.
  • Series divergence means no limit is reached.
  • Divergent series may increase indefinitely.
  • Stopping at a point is impossible, due to continuous addition.
Constant Series
A constant series is one where each term is exactly the same. Imagine watching a movie that repeats the same scene continuously. Here, the constant refers to each repeating scene. In the exercise provided, the series \( \sum \frac{11}{1+100^{-2}} \)is constant because each term is the same value, \( \frac{11}{1.01} \).

Constant series have unique properties:
  • If the constant term is zero, the series is zero everywhere, which means it converges as its sum is constant and finite (zero).
  • If the constant is anything other than zero, adding the same number an infinite number of times leads to divergence because the series will keep growing larger and larger.
This constancy can simplify our analysis by letting us focus on the nature of the highly predictable pattern.
Convergence Test
To determine whether a series converges or diverges, we use convergence tests. A convergence test is like a mathematical toolkit that helps us explore whether our series will settle down to a particular value.

For constant series, there's a simple rule: 鈥� If each term value is zero, the series converges. 鈥� If the value is non-zero, it diverges.
Let's say we have a series where each term is added up to infinity. We apply a test to see if the entire added value stays under control. If the test says the sum doesn鈥檛 grow indefinitely, it converges.
  • Helpful for deciding series behavior.
  • Zero terms mean finite sum (convergence).
  • Non-zero constant means unlimited growth (divergence).
In practice, these tests make sense of infinite sums, similar to checking if a car keeps moving or eventually parks at a destination.

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