Infinite Series Question, difference between infinities Was just thinking about infinite series, what is wrong with the below?
$$s_1=\lim _{n\to\infty} \sum_{i=1}^n \frac{1}i =\infty $$
$$s_2=\lim _{n\to\infty} \sum_{i=1}^n i =\infty $$
$$\therefore \ s_1=s_2$$
Fixed the formula, I am not sure if this is wrong but every element of $s_2$ is larger than the corresponding element of $s_1$ (apart from i=1), is this just an oddity of infinity?
 A: The two series that you write diverges, so there is no real number that these sums approach as you take the limit. Sometimes one uses the symbol $\infty$ to denote the value of these sums, and in this sense, the two sums "converges" to the same limit -infinity. In my opinion, this is meaningless since this series diverges.
The example that you gave is good to see the differences between the infinities. The two sums have very different "rates of divergence". To see this, consider the partial sums
\begin{align}
a_n = \sum_{k=1}^{n} \frac{1}{k} \; , \\[6pt]
b_n = \sum_{k=1}^{n} k \; .
\end{align}
The sum of the first $n$ positive natural number is well known to give
$$ b_n = \frac{n(n+1)}{2} \; . $$
Using the integral test, we can see that the sum of the Harmonic numbers is in the limit
$$ \ln n  = \int_1^{n} \frac{1}{k} dk \; \leq a_n \leq  \; 1 +  \int_1^{n} \frac{1}{k} dk = 1 + \ln n \; . $$
Thus, for large $n$ the partial sums behave like
\begin{align}
a_n  \; &\sim \; \ln n \; , \\
b_n \;&  \sim \; \frac{1}{2} \, n^2 \; , 
\end{align}
which is enough to see that the two series diverge. However, more importantly, we can see that the Harmonic series diverges very slowly in a logarithmic way were as the sum of the natural number diverges quadratically.

To answer the question, there is nothing wrong with it,  it is just notation. I just wanted to explain that you are right when you compare one partial sum with the other and they have grown very differently. For finite $n$ we can easily see that $a_n < b_n$ (and even $ \text{Exp} \, a_n < b_n$). However, when you sum over infinite number $n\in \mathbb{N}$ the two sums goes to infinity. After you sum the information on the rate of grown of the function is lost and there is no meaning in "$\infty < \infty$", both diverges (are $\infty$).
