Series

In this post we will continue to investigate more theorems of convergence tests. Tests D’Alembert Ratio Test $\textbf{\small Theorem 1.17.:} $ Let $\sum_{n=1}^{\infty} a_n$ be positive series and let $l = \lim_{n\rightarrow \infty}\frac{a_{n+1}}{a_n}$. Then, series converges if $l < 1$, series diverges if $l > 1$, and gives no information if $l = 1$ or limit fails to exist. $\textbf{\small Proof:} $ Let $l < 1$. We want to prove that $\sum_{n=1}^{\infty} a_n$ converges. Let us write what we know from assumption. Choose $\varepsilon > 0$ and $\forall n > N$ and $0 < (l + \varepsilon) < 1$: ...

Telescoping Series In this post, we will learn more about infinite series. We shall start with so-called telescoping series, that has a form of $$ \begin{aligned} \sum_{k=1}^{\infty} (a_n - a_{n+1}) \end{aligned} $$ $$ \begin{aligned} \sum_{n=1}^{\infty} (a_n - a_{n+1}) \end{aligned} $$ Let us investigate this sum step by step. Observe that the partial sum $s_n$ can be written as $$ \begin{aligned} s_n = a_1 - a_2 + a_2 - a_3 + \dots + a_n - a_{n+1} \end{aligned} $$ $$ \begin{aligned} s_n = a_1 - a_2 + a_2 - a_3 + \dots + a_n - a_{n+1} \end{aligned} $$ So we can just simplify this ...

By the intuition from our birth, we love summing things, categorising the similar things and so on. Consider a sum $A=1+2+3+4+\dots$ what we have here is infinite sum, as “$\dots$” imply “this sum goes to infinity with the order you see”. So in a some sense, it behaves like sequences, because we write the things in order like $1$, $2$, $3$, $4$, $\dots$ and then we sum them. Mathematicians tought that so, and as a result they came up with a new topic called “series”, especially “infinite series”. Spoiler alert, an infinite series can converge, if not we say series is divergent. ...