Problem 40 - Calculus

Question

By recognizing each series as a Taylor series evaluated at a particular value of \(x,\) find the sum of each of the following convergent series. $$1-0.1+0.1^{2}-0.1^{3}+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \(\frac{10}{11}.\)

Step 1: Identify the Taylor Series

The given series \(1 - 0.1 + 0.1^2 - 0.1^3 + \cdots\) is an infinite geometric series. Let's recognize it as a Taylor series. We can write it as \(\sum_{n=0}^{\infty} (-0.1)^n.\) This resembles the Taylor series for \( \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \), evaluated at \(x = -0.1.\)

Step 2: Verify the Convergence Condition

For the Taylor series of \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \) to be valid, \(x\) must satisfy \(|x| < 1\). In our series, \(x = -0.1\), and \(|-0.1| = 0.1 < 1\), so the series converges according to this condition.

Step 3: Find the Sum Using the Taylor Series Formula

Since the series converges and is in the form of the geometric series \(\sum_{n=0}^{\infty} (-0.1)^n\), we can find its sum using the formula for the sum of a geometric series. The sum of \( \sum_{n=0}^{\infty} x^n \) is \( \frac{1}{1-x} \). Substituting \(x = -0.1\) gives us \( \frac{1}{1 - (-0.1)} = \frac{1}{1 + 0.1} = \frac{1}{1.1}.\)

Step 4: Simplify the Sum

Calculate the expression \( \frac{1}{1.1} \). This simplifies to \( \frac{10}{11} \). So, the sum of the series is \( \frac{10}{11}.\)

Key Concepts

Geometric Series

Let's start with what a geometric series is. A geometric series is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In math terms, a geometric series can be expressed as:

• First term: \( a \)
• Common ratio: \( r \)
• Series: \( a + ar + ar^2 + ar^3 + \ldots \)

In the original exercise, we had the series: \(1 - 0.1 + 0.1^2 - 0.1^3 + \cdots \). Here, the first term \(a\) is 1, and the common ratio \(r\) is \(-0.1\). Each subsequent term is obtained by multiplying the previous one by \(-0.1\). Recognizing this pattern is crucial when dealing with series since it allows us to apply specific formulas to simplify and solve them.

Convergence

Convergence is a key concept when dealing with infinite series like geometric series. An infinite series converges if the sequence of its partial sums approaches a specific number, called the limit. For geometric series, convergence mainly depends on the absolute value of the common ratio \(|r|\).
The rule is simple:

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

In our series, the common ratio \(r\) is \(-0.1\), and since \(|-0.1| = 0.1\), which is less than 1, the series converges. This is a critical step to determine whether or not we can calculate the sum using the series formula. By establishing convergence, we're able to proceed with further calculations confidently.

Series Sum Calculation

Once we know a series converges, we can find its sum. For geometric series, the sum can be computed using the formula for an infinite series:
\[ S = \frac{a}{1-r} \]where \(a\) is the first term and \(r\) is the common ratio.
In our specific example:

• \(a = 1\)
• \(r = -0.1\)

Substituting these values into the formula, we calculate:
\[ S = \frac{1}{1 - (-0.1)} = \frac{1}{1.1} \]
This simplifies to \(\frac{10}{11}\). Understanding these calculations is fundamental, as it shows us how the infinite sum can be reduced to a simple fraction, making it much easier to handle.