Final answer:
A telescoping series is a series where most of the terms cancel out, leaving behind a simple expression that can be evaluated easily. To find the sum of a telescoping series, we need to look for a pattern in the terms and simplify the expression. In this case, the subject of the question is Series and Sequences (option a).
Step-by-step explanation:
A telescoping series is a series where most of the terms cancel out, leaving behind a simple expression that can be evaluated easily. In order to find the sum of a telescoping series, we need to look for a pattern in the terms and simplify the expression. Let's consider an example:
Example: Find the sum of the series S = 1/2 + 1/4 + 1/8 + 1/16 + ...
Step 1: Notice that each term is half of the previous term. We can write the terms as 1/2^1, 1/2^2, 1/2^3, 1/2^4, ...
Step 2: Let's express the series in terms of a common denominator. The denominator for each term is a power of 2. We know that any number can be expressed as a power of 2, so let's use a variable n to represent the power: 1/2^n.
Step 3: Now, let's try to express the sum of the series in a simplified form. We can rewrite the series as S = 1/2^1 + 1/2^2 + 1/2^3 + 1/2^4 + ...
Step 4: Notice that if we subtract each term from the next term, most of the terms will cancel out. Let's subtract each term from the next term: 1/2^1 - 1/2^2 + 1/2^2 - 1/2^3 + 1/2^3 - 1/2^4 + ...
Step 5: We can see that all the terms cancel out except the first term (1/2^1) and the last term (-1/2^n). Therefore, the sum of the series is simply the first term minus the last term. The last term is -1/2^n, so the sum of the series is S = 1/2^1 - (-1/2^n) = 1/2 - (-1/2^n).
This is how we find the sum of a telescoping series. Now, let's look at the options given. The subject of this question is Series and Sequences (option a).