Sum coefficients

Question

Assume this equailty: $$\sum_{n=1}^\infty a_n = \sum_{n=1}^\infty b_n$$ Does $$a_n = b_n$$ hold for every $a_n$ and $b_n$?

Answer 1

$a_1=1, a_n=0$ for $n\geq 2$ and $b_1=b_2=\frac{1}{2}, b_n=0$ for $n\geq 3$ is a counterexample.

Answer 2

Let consider as counter example any convergent series such that $a_n=b_{n+1}$ with $b_1=0$ with $b_n>0$ strictly decreasing for $n>1$.

---

Therefore we can claim that for any convergent series $\sum a_n$

• $a_n=b_n \Rightarrow \sum a_n=\sum b_n$ (trivial)

but

• $\sum a_n=\sum b_n \not \Rightarrow a_n=b_n $

Note that the latter doesn't mean that the implication

• $\sum a_n=\sum b_n \Rightarrow a_n=b_n $

is always false but that some case exist for which it is false, as all the counter examples presented here have shown.

Answer 3

There are of course, more than enough counter-examples to the statement mentioned in the post. This should intuitively be clear too, because you have only one condition $$\displaystyle\sum_{n=1}^\infty a_n = \displaystyle\sum_{n=1}^\infty b_n$$ but infinitely many statements to prove $$a_n=b_n \ \forall n\in \mathbb{N} \tag{1}$$

However, $(1)$ would be true ;-) if you were given $$\displaystyle\sum_{n=1}^k a_n = \displaystyle\sum_{n=1}^k b_n \ \forall k\in \mathbb{N}$$