# Tagged Questions

2answers
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### What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
1answer
187 views

### Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?

Q1: Can we prove that all zeros of cos(x) are real from the following Taylor series expansion of cos(x)? $$\cos(x) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k}$$ The Riemann $\xi(z)$ function is ...
2answers
361 views

### A Question On Euler's Proof Of the Basel Problem

I've studied the proof that Euler gave for the famous Basel Problem, and it would seem that while it is technically correct, he does not justify all of his steps properly. Namely, he assumes that ...
0answers
171 views

### Expanding an expression using Taylor's series

We've been attempting to expand an expression with Taylor's Theorem but can't quite make the math work out.  \frac{f\left(x_n\right)}{f'\left(x_n\right)}= \frac{1}{m}\frac{f^{(m)}\left(\xi ...
2answers
432 views

### Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...