- #1
Saitama
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Evaluate the following:
$$\Large \sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k}$$
$$\Large \sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k}$$
Opalg said:[sp]$$\begin{aligned}\sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k} &= 1 + \frac12 - \frac13 - \frac14 + \frac15 + \frac16 - \frac17 -\frac18 + \ldots \\ &= \Bigl(1 - \frac13 + \frac15 - \frac17 + \ldots\Bigr) + \frac12\Bigl(1 - \frac12 + \frac13 - \frac14 + \ldots\Bigr) \\ &= \frac\pi4 + \frac12\ln2\end{aligned}$$[/sp]
Summation is a mathematical operation that adds together a sequence of numbers. It is represented by the symbol ∑ (sigma) and is commonly used in various fields of science, such as physics, statistics, and computer science.
A summation challenge is a problem that requires you to evaluate a given summation expression. It is designed to test your understanding of summation notation and your ability to perform the necessary calculations.
To evaluate a summation expression, you need to first identify the starting and ending values of the summation, as well as the function or sequence being summed. Then, you can plug in the values and use mathematical operations, such as addition or multiplication, to calculate the final result.
Some common types of summation expressions include arithmetic series, geometric series, and telescoping series. Each type has its own formula for evaluating the summation, so it's important to familiarize yourself with these formulas before attempting to solve a summation challenge.
Some tips for solving summation challenges include carefully reading the given expression, breaking down the problem into smaller parts, and using known summation formulas. It's also helpful to check your work and use a calculator when necessary to avoid errors.