Question

If \(|\mathrm{x}|<1\), then the sum of the infinite series \(\left[\mathrm{x}+\frac{1}{2}\right]+\left[\mathrm{x}^{2}+\mathrm{x} \frac{1}{2}+\left(\frac{1}{2}\right)^{2}\right]+\left[\mathrm{x}^{3}+\mathrm{x}^{2} \frac{1}{2}+\mathrm{x}\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3}\right]+\ldots . \infty\) (a) \(\frac{1}{2}+x\) (b) \(\frac{2-x}{1-x}\) (c) \(\frac{1+x}{1-x}\) (d) \(\frac{2+x}{2-x}\)

Short answer

Answer: (b) \(\frac{2-x}{1-x}\)

Step-by-step solution

Write out the series explicitly

We'll write out the first few terms of the series to make it easier to manipulate and find patterns: \(S = \left[x + \frac{1}{2}\right] + \left[x^2 + \frac{1}{2}x + \frac{1}{4}\right] + \left[x^3 + \frac{1}{2}x^2 + \frac{1}{4}x + \frac{1}{8}\right] +\ldots\)

Rearrange the terms of the series

We can regroup the terms in the series based on the powers of x: \(S = \left[x + x^2 + x^3 +\ldots\right] + \left[\frac{1}{2} +\frac{1}{2}x + \frac{1}{2}x^2 + \ldots\right] + \left[\frac{1}{4}+\frac{1}{4}x+\ldots\right]+ \ldots\)

Identify that each group is a geometric series

Each group in the rearranged series is a geometric series with the first term \(a\) and the common ratio \(r = x\): 1. The first group: \(a = x\), 2. The second group: \(a = \frac{1}{2}\), 3. The third group: \(a = \frac{1}{4}\), and so on.

Find the sum for each geometric series

As \(|x| < 1\), the sum of each geometric series can be found using the formula \(S_n = \frac{a}{1 - r}\). We will now find the sum for each of the groups: 1. The first group: \(S_1 = \frac{x}{1-x}\) 2. The second group: \(S_2 = \frac{\frac{1}{2}}{1-x} = \frac{1}{2(1-x)}\) 3. The third group: \(S_3 = \frac{\frac{1}{4}}{1-x} = \frac{1}{4(1-x)}\) and so on.

Sum all the group sums to find the total sum

Now, to find the total sum of the infinite series, we will sum all these group sums. \(S = \frac{x}{1-x} + \frac{1}{2(1-x)} + \frac{1}{4(1-x)} +\ldots\) Notice that this is another geometric series with \(a = \frac{x}{1-x}\) and \(r = \frac{1}{2}\). Since \(|\frac{1}{2}| < 1\), we can again use the formula \(S_n = \frac{a}{1 - r}\): \(S = \frac{\frac{x}{1-x}}{1-\frac{1}{2}}\)

Simplify the final sum

Now, we'll simplify the expression for the sum of the infinite series: \(S = \frac{\frac{x}{1-x}}{\frac{1}{2}} = \frac{2x}{1-x}\) Comparing the result with the choices given, we find that the correct answer is (b) \(\frac{2-x}{1-x}\).

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Geometric series can be finite or infinite, depending on how many terms they include.
The general formula for a geometric series is:

• First term: \( a \)
• Common ratio: \( r \)

For an infinite geometric series, the sum \( S \) of the series can be calculated using the formula: \[ S = \frac{a}{1 - r} \]This formula is valid if the absolute value of the common ratio \( |r| \) is less than 1.
If the absolute value of \( r \) is equal to or greater than 1, the series does not converge, meaning that it does not sum to a finite value.

Sum of Infinite Series

The sum of an infinite series can be determined for cases where the series converges, that is, it approaches a specific limit as the number of terms grows indefinitely large.
It's particularly useful when dealing with geometric series where the ratio \( |r| < 1 \).

When applied to the geometric series, each component or group within the series contributes to a part that sums up according to the same type of geometric principles. In the exercise provided, each group consists of terms that follow a geometric series pattern, for which the sum is calculated independently before summing up the entire series.

• The first part is summed using the ratio and initial term of that specific sub-series.
• The series needs careful observation of the pattern it follows, and typical manipulation involves rearranging terms into identifiable geometric segments.

Once these separate sums are evaluated, they contribute to the total sum of the infinite series, effectively capturing the behavior of the whole sequence.

Algebraic Manipulation

In solving series problems, algebraic manipulation often comes into play to simplify and rearrange terms.
By restructuring the way terms are arranged, especially using patterns, we can see new ways to sum or manage complex relationships between numbers.
For instance, in an infinite series, after breaking it into components or groups, we can treat each segment as a standalone geometric series.
To achieve this:

• Recognize patterns within the series.
• Separate terms based on these identified patterns.
• Reapply known formulae, like that of the geometric series, effectively.

In our example, after isolating the geometric series segments, we apply the sum formula iteratively and simplify it to understand the overall sum.
Such processes help transform complex infinite series into algebraically manageable forms, enabling straightforward solutions.