Question

Does the sum of a convergent and divergent series always convergeee? My textbook says nothing about this

Answer 1

the sum of a convergent and divergent series never converges

Answer 2

we can prove it

Answer 3

Suppose you are given a convergent series a_n and a divergent series b_n.
If the sum exists, call it c.
$$
\Large
{ \sum a_n + \sum b_n = c
\\
\sum b_n = c - \sum a_n
}
$$

Answer 4

now you have a contradiction

Answer 5

we can also write it like this.
Suppose that
Convergent series + divergent series = k , for some constant k

subtract both sides the convergent series

divergent series = k - convergent series

but the right side is a finite number, since its k - a
and the left side is divergent

Answer 6

Ah, thank yoooou.

Answer 7

This might be clearer.
$$
\Large
{ \sf suppose~ that \\
\sum a_n = k, \sum b_n = does~ not~ exist
\\
If ~~\sum a_n + \sum b_n = C
\\~\\ then \\~\\
\sum b_n = C - \sum a_n
\\ \sum b_n = C - A
\\~\\ then \\~\\ \sum b_n~ does ~exist
\\ ~\\ contradiction! \\
}

$$

Answer 8

Thank you for the fast response. =)