Prove that the convergence or divergence of series is not affected by multiplying each term by the same non-zero constant

Suppose that we have a converging series like:

(a₁ + a₂ + a₃ + ... ) = A

or

∑aₙ = A

So this series converges to the value A.

Now, let's multiply all the coefficients by a real number c, different than zero, then the sum is:

(c*a₁ + c*a₂ + c*a₃ + ...)

now we can take the common factor c out, to get:

c*(a₁ + a₂ + a₃ + ...)

And the thing inside the parentheses is equal to A, then we have:

c*(a₁ + a₂ + a₃ + ...) = c*A

So the series still converges for any value of c.

Now, suppose that this series diverges (to minus infinity or infinity)

Then we have:

(b₁ + b₂ + b₃ + ...) = ∞

Same as before, we multiply all the terms by the same number c:

(c*b₁ + c*b₂ + c*b₃ + ...)

again, we take the common factor out:

c*(b₁ + b₂ + b₃ + ...)

And same as before, we get:

c*(b₁ + b₂ + b₃ + ...) = c*∞

and c*∞ is still equal to ∞ (or -∞ if c is negative), so the series still diverges.

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