Difference between equals/approaches/approximate

Question

Consider the series

$$\sum\limits_{k=0}^{\infty} \frac{1}{2^k} = 2$$

Is it correct to say "$\text{the series approaches 2 ?}$" if so, shouldn't we replace $=$ with $\approx$ ?

Also Is it correct to say "$\text{the sequence of partial sums approaches 2?}$"

Answer 1

No. In general the $\approx$ sign is not used in mathematics except to indicate that an equality is not exact - which tends to be, "This big awful computation is near this decimal" and the notation is purely for convenience of the reader, since it lacks precise meaning. (Of course, this doesn't mean that no one ever gives $\approx$ a precise meaning and uses it accordingly - but this is not a standard use that would be clear without explanation)

It happens that the left side is a limit. A limit is a thing that can be evaluated. The statement

$a_n$ approaches $P$

is equivalent to

$$\lim_{n\rightarrow\infty}a_n=P$$

and we don't need to introduce any new notation other than equality because a limit represents a definite value (if one exists). So, when we write $$\sum_{k=0}^{\infty}\frac{1}{2^k}=2$$ we really mean "the left side equals the right side", but this is easily known to be equivalent to, "the sequence of partial sums approaches 2".

It's the same as how we might write $$\sin(\pi)=0$$ rather than have to introduce a new notation for "The sine function vanishes at $\pi$".

If we were really intent on thinking about approximations, something like $$a_{1000}\approx P$$ would not be unreasonable if were true, and we had some compelling reason to say it - for instance, if we have some easy to compute series converging to some constant, like $\pi$, we might use the above to indicate that we are using the $1000^{th}$ term of the series to stand in for the real value (which we can't so easily manipulate computationally).

Answer 2

The symbol $\sum_{k=0}^\infty a_k$ has two different meanings: The first one is, that $\sum_{k=0}^\infty a_k$ is a shortcut for the sequence of partial sums, i.e. $\sum_{k=0}^\infty a_k:=\left(\sum_{k=0}^n a_k\right)_n$. One the other hand, it also means the limit of this sequence (if it exists), i.e. $\sum_{k=0}^\infty a_k:=\lim_{n\to\infty}\sum_{k=0}^na_k$. In your case, this limit is equal to 2, so in this sense, the equal sign does indeed make sense.