The properties of convergent series

Scalar multiplication

Let \( (a_n) \) be a numerical sequence and \(\lambda \in \mathbb{R}\) a real number.

Suppose that the series \(\sum a_n\) converges to a limit \(l\), which is expressed by the limit of its partial sums:

Let us consider the sequence of partial sums of the series \(\sum (\lambda \ a_n)\). For any integer \(n \geqslant t\), the linearity properties of finite sums allow us to factor out the scalar coefficient:

Taking the limit as \(n \to +\infty\) and applying the general limit laws, we obtain the following chain of equalities:

The sequence of partial sums admits a finite limit equal to \(\lambda \ l\), which proves the convergence of the series.

Thus, finally:

Addition

Let be \( \bigl(a_n, \ b_n \bigr) \) two numerical sequences.

1. If both series converge

   In the same way as before, if \(\sum a_n\) and \(\sum b_n\) converge, then:

   $$ \exists! \ (l, l') \in \hspace{0.04em} \mathbb{R}^2, \ \left \{ \begin{gather*} \lim_{n \to + \infty} \sum_{k = t}^{n} a_k = l \\ \\ \lim_{n \to + \infty}\sum_{k = t}^{n} b_k = l' \end{gather*} \right \} $$

   So by adding the two expressions, we have:

   $$ \lim_{n \to + \infty} \sum_{k = t}^{n} a_k + \lim_{n \to + \infty} \sum_{k = t}^{n} b_k = l + l' $$

   The limit of a sum being the sum of the limits:

   $$ \forall (f, \ g),$$

   $$\lim (f+g) =\lim (f) +\lim (g) $$

   So now we have this:

   $$ \lim_{n \to + \infty} \biggl[ \sum_{k = t}^{n} a_k + \sum_{k = t}^{n} b_k \biggr] = l + l' $$
   $$ \lim_{n \to + \infty} \biggl[ \sum_{k = t}^{n} (a_k + b_k) \biggr] = l + l' $$

   Therefore, we obtain that \(\sum (a_k + b_k) \) converge towards \((l + l')\).

   
   

   So,

   $$ \forall \bigl(a_n, \ b_n\bigr),$$
   $$ \sum a_n \text{ and } \sum b_n \text{ converges } \Longrightarrow \sum (a_n + b_n) \text{ converges } $$

   $$ \Longrightarrow $$

   $$ \sum_{k = t}^{+ \infty} (a_k + b_k) = \sum_{k = t}^{+ \infty} a_k + \sum_{k = t}^{+ \infty} b_k $$

2. If one of the series diverge

   Suppose that the series \(\sum a_n\) converges to a real number \(l\) and that the series \(\sum b_n\) diverges.

   By contradiction, if the sum series \(\sum (a_n + b_n)\) were convergent and tended toward a limit \(L\), then by linearity of summation, the following difference series should also converge to \(L - l\):

   $$ \sum_{k=t}^n b_k = \sum_{k=t}^n \bigl[ (a_k + b_k) - a_k \bigr] $$

   Taking the limit of the partial sums, we would obtain:

   $$ \lim_{n \to +\infty} \sum_{k=t}^n b_k = \lim_{n \to +\infty} \sum_{k=t}^n (a_k + b_k) - \lim_{n \to +\infty} \sum_{k=t}^n a_k = L - l $$

   Since the right-hand side is a fixed real number (\(L - l\)), the sequence of partial sums of \(b_n\) would admit a finite limit, meaning that the series \(\sum b_n\) would converge.

   This directly contradicts our initial hypothesis. Therefore:

   
   

   Thus,

   $$ \forall \bigl(a_n, \ b_n\bigr),$$
   $$ \sum a_n \text{ converges and } \sum b_n \text{ diverges } \Longrightarrow \sum (a_n + b_n) \text{ diverges } $$

Identification of series having the same nature

Let \((c_n, d_n)\) be two sequences having strictly positive terms from the index \(t\), and \(l \in \mathbb{R}^*_+ \) a real number such as:

Invoking the definition of a limit, it is possible to find two positives real numbers \((l_1, l_2) \in (\mathbb{R}_+)^2\) such as:

By multiplying all terms by \(d_n\), we do obtain this:

1. If \(\sum d_n\) converge

   If \(\sum d_n\) converge, then it is the same thing for \(\sum (l_2 \ d_n)\).

   In addition to that, the series \(\sum c_n\) being lower than a convergent series, it also converges.

   $$ \lim_{n \to + \infty} \left[ \frac{c_n}{d_n} \right] = l > 0 \Longrightarrow \text{\(\sum d_n\) converge \(\Longrightarrow \sum c_n\) converge} $$

   Note :

   If both series have strictly negative terms, inequalities direction change:

   $$ \sum_{k=t}^n (l_1 \ d_k) > \sum_{k=t}^n c_k > \sum_{k=t}^n (l_2 \ d_k) $$

   But the series \(\sum c_n\) is still trapped by \(\sum d_n\), and the result is the same.

2. If \(\sum d_n\) diverge

   Likewise, if \(\sum d_n\) diverge, then it is the same thing for \(\sum (l_1 \ d_n)\).

   In the same way, the series \(\sum c_n\) being greater than a divergent series, it also diverges.

   $$ \lim_{n \to + \infty} \left[ \frac{c_n}{d_n} \right] = l > 0 \Longrightarrow \text{\(\sum d_n\) diverge \(\Longrightarrow \sum c_n\) diverge} $$

   Remarque :

   If both series have strictly negative terms, the result is still the same for the same reasons.

And as a result,

For any couples of series \(\bigl(c_n, \ d_n\bigr)\) habing an indentical and constant sign from a certain index:

Moreover, if \((l = 1)\) both sequences are equivalent :

We will have as a bonus: