Question

Use the properties of infinite series to evaluate the following series. $$\sum_{k=0}^{\infty}\left[\frac{1}{2}(0.2)^{k}+\frac{3}{2}(0.8)^{k}\right]$$

Short answer

Answer: The sum of the given infinite series is $$\frac{65}{8}$$.

Step-by-step solution

Sum of the first geometric series

To find the sum of the first geometric series $$\frac{1}{2}(0.2)^{k}$$, we use the sum formula $$S = \frac{a}{1-r}$$ and substitute a1 = $$\frac{1}{2}$$ and r1 = 0.2: $$S_{1} = \frac{\frac{1}{2}}{1 - 0.2}$$

Calculate the sum of the first geometric series

Now, perform the arithmetic to find the sum of the first geometric series: $$S_{1} = \frac{\frac{1}{2}}{0.8} = \frac{5}{8}$$

Sum of the second geometric series

Next, find the sum of the second geometric series $$\frac{3}{2}(0.8)^{k}$$ using the same sum formula. Substitute a2 = $$\frac{3}{2}$$ and r2 = 0.8: $$S_{2} = \frac{\frac{3}{2}}{1 - 0.8}$$

Calculate the sum of the second geometric series

Perform the arithmetic to find the sum of the second geometric series: $$S_{2} = \frac{\frac{3}{2}}{0.2} = \frac{15}{2}$$

Add the sums of both geometric series

Now, add the sums of both geometric series to find the sum of the given infinite series: $$\sum_{k=0}^{\infty}\left[\frac{1}{2}(0.2)^{k}+\frac{3}{2}(0.8)^{k}\right] = S_{1} + S_{2} = \frac{5}{8} + \frac{15}{2}$$

Calculate the final sum

Perform the arithmetic to find the sum of the given infinite series: $$\sum_{k=0}^{\infty}\left[\frac{1}{2}(0.2)^{k}+\frac{3}{2}(0.8)^{k}\right] = \frac{5}{8} + \frac{15}{2} = \frac{5+60}{8} = \frac{65}{8}$$ Therefore, the sum of the given infinite series is $$\frac{65}{8}$$.

Key concepts

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. Think of it as continuing endlessly without stopping. We often describe these using the summation symbol \( \sum \) which stands for adding together all the elements from a sequence. In practical problems, only certain types of infinite series can be summed to get a meaningful result.
A straightforward example is a geometric series, where each term is a fixed multiple of the previous term. By understanding these patterns, we can sometimes find the complete sum, even though the series is infinite.

Sum of Series Formula

The sum of a geometric series formula is a handy formula. It lets you find the sum of all terms when the sequence goes on forever. For a geometric series to sum up, the common ratio \( r \) must be between -1 and 1. It's written as \( S = \frac{a}{1-r} \), where:

• \( a \) is the first term of the series.
• \( r \) is the common ratio between terms.

This formula works because the terms get smaller and smaller, approaching zero, which is why we can calculate a total sum. In our exercise, we used this to determine the sum for two separate series parts and then added them together.

Arithmetic Operations

Arithmetic operations are the basic math functions: addition, subtraction, multiplication, and division. These operations help us manipulate numbers to find answers. When working out the sum of series like in our exercise, arithmetic operations were essential.
First, we calculated the individual sums for each geometric series. For the first series, division was used to find \( S_1 \) as \( \frac{5}{8} \). Then, the second series' sum \( S_2 \) was calculated as \( \frac{15}{2} \). Finally, both sums were added together using the addition operation to arrive at the total sum of \( \frac{65}{8} \). Arithmetic steps make all the mathematical calculations clear and accessible.

Series Convergence

When we study series, one important aspect is convergence. A series converges when it approaches a specific number as more terms are added. In simpler terms, the series settles to a definite value, rather than growing endlessly or oscillating.
For geometric series, whether they converge or not depends on the common ratio \( r \). To converge, \( r \) must be less than 1 in magnitude. That's why in our exercise, the first series' ratio \( 0.2 \) and the second series' ratio \( 0.8 \) both allow their respective series to converge.

• It converges because each term becomes tinier as it progresses.
• Ultimately, the sum approaches a finite limit which we calculated using the sum of series formula.

Convergent series are crucial in maths because they guarantee a tangible result.