What is the connection between mathematical induction and implication? I always see that mathematical induction is about
P(k)->P(k+1).
From what I know, mathematical induction works by finding a way to transform P(k) into P(k+1) and if you can do it, then you prove it.
Now, speaking in terms of logic formality, we can treat P(k) as statement A, and treat P(k+1) as statement B. We know that an implication is true if statement A is true and Statement B is true; A is false and B is true/false (implication is false if A is true and B is false).
I tried looking on Google, but there are some fairly contradictory results.
I thought I'd ask you guys so we could get an authoritative answer on the subject!
I would like to evaluate the sum
$$
\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)
$$
Where $\operatorname{Si}$ is the sine integral, defined as:
$$\operatorname{Si}(x) := \int_0^x \frac{\sin t}{t}\, dt$$
I found that the sum could be also written as
$$
-\sum\limits...
