It is useful to write series both in the form \(a_{1}+a_{2}+a_{3}+\cdots\) and in the form \(\sum_{n=1}^{\infty} a_{n} .\) Write out several terms of the following series (that is, write them in the first form). $$ \sum_{n=1}^{\infty} \frac{\sqrt{n}}{n+1} $$

When dealing with sequences and series, we often encounter a special notation: the summation notation or series notation. This is represented by the sigma symbol \(\backslash sum\). For example, \[ \backslash sum_{n=1}^{\backslash infty} \backslash frac{\backslash sqrt{n}}{n+1} \] tells us to sum up the terms given by the formula \(\backslash frac{\backslash sqrt{n}}{n+1}\) starting from \ n = 1\ all the way to infinity. This concise way of writing allows us to represent long or even endless sequences succinctly.
Series notation is especially useful for infinite series, where we sum infinitely many terms. Instead of writing out each term, the sigma notation helps us see the pattern clearly and saves space.
For example: \[ \backslash sum_{n=1}^{\backslash infty} a_n = a_1 + a_2 + a_3 + \backslash cdots \] Here, \ a_n \ stands for the general term of the series.

The general term is crucial when working with series. It represents the formula that generates the terms in the sequence. In our example, the general term \ a_n \ is \[ a_n = \backslash frac{\backslash sqrt{n}}{n+1} \] This means each term in the series is found by plugging in successive values of \ n \ into the formula.
For instance, when \ n = 1 \, the term is: \[ a_1 = \backslash frac{\backslash sqrt{1}}{1+1} = \backslash frac{1}{2} \] For \ n = 2 \: \[ a_2 = \backslash frac{\backslash sqrt{2}}{2+1} = \backslash frac{\backslash sqrt{2}}{3} \] And so on. By identifying the general term, we can generate any term in the series and understand the pattern it follows.

Summation is the process of adding up all the terms in a series. When we write \ \backslash sum_{n=1}^{\backslash infty} \backslash frac{\backslash sqrt{n}}{n+1} \ we are indicating that we want to find the sum of all terms from \ n = 1 \ to infinity.
To understand this better, we can start by writing out the first few terms as shown in the solution: \[ \backslash frac{1}{2} + \backslash frac{\backslash sqrt{2}}{3} + \backslash frac{\backslash sqrt{3}}{4} + \backslash frac{2}{5} + \backslash cdots \] Even though it's an infinite series, these initial terms give us a clearer picture of what we are summing. Summation in series can help determine convergence, where we check if the total sum approaches a finite value as we add infinitely many terms. Understanding this process is key to mastering series in mathematics.