comment Show the closed form of the sum $\sum_{i=0}^{n-1} i x^i$
Why not prove it by induction?
comment I can't find the option none for payment verification
See also apple.stackexchange.com/a/163532/6824
comment What happens to unaired or one time only aired pilots?
I remember ... sometime in the 60's I guess: One of the US networks during the summer ran a whole series of failed pilots. I remember thinking that some of them were quite good.
comment Hahn-Banach Thm for Normed Space
Show your thoughts. Since you know the Hahn-Banach theorem, tell us why you cannot simply cite that. Also, figure out the misprint in your statement.
revised Hahn-Banach Thm for Normed Space
comment How do I evaluate the Wronskian for this equation
Note: that's $W(1)$ we know, $W(0)$ is undefined.
comment Prove the Maclaurin-series representation of $ \sqrt{1 + x} $.
For part (b) you may need a little care to evaluate at the endpoint of the interval of convergence.
comment Does Limit exist or not?
Needs more context. That limit does not exist in the real numbers with its usual topology. That limit does exist in the Riemann sphere with its usual topology.
comment If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?
In other words, observe that the inequality in Fatou's Lemma can be strict.
comment Prove that topology over rationals is different
Can you find a set that is open in this new topology that is not open in the standard topology?
comment Maesurability of function $\mathbb R\ni t \mapsto P(X>t)$
The function $t \mapsto P(X>t)$ is decreasing (possibly not strictly). So it is certainly Borel measurable. It is also Riemann integrable. Similarly $t \mapsto P(x \ge t)$ is Borel measurable. So the difference $t \mapsto P(X=t)$ is Borel measurable.
comment Why did Glinda say she was the Good Witch of the North in The Wizard of Oz (1939)
In the book The Wonderful Wizard of Oz (1900) she is the Good Witch of the South. So maybe the question is why that was changed for the movie in 1939...
comment Shouldn't 'dawn' come before 'rise'?
Maybe "rise" has meanings other than sunrise...
comment What are the types of elements in the set $\mathbb{Q} + i \mathbb{Q}$?
In addition to what you say you already know, you will need to show that $i\mathbb Q$ is countable... can you do that?
comment What are the types of elements in the set $\mathbb{Q} + i \mathbb{Q}$?
$\mathbb Q \times i\mathbb Q$.
comment Is the following a valid characterisation of complete metric spaces?
Hints: (1) Show that a Cauchy sequence has a subsequence with the $r^{-i}$ property you mention. (2) If a Cauchy sequence has a convergent subsequence, then the whole sequence converges.
comment solving the integral of $e^{x^2}$
"No closed form" is too vague. Better to say, "not an elementary function".
comment Paper on the Impact of Science and/or Math on American History
Maybe you need to confine yourself to one item. Say the mathematics that makes a CT scan possible: reconstructing a 3-D model from many 2-D x-ray images. en.wikipedia.org/wiki/Radon_transform
comment Unknown 80s movie with closet-portal & goat-legged man
Certainly mention of a portal in a closet leading to another world reminded me of "Time Bandits". But I don't remember a goat-legged man. There is a bull-headed man, though.
comment Does Pop-Science involvement hurt one's reputation?
Is this "pop science" done instead of peer-reviewed research? Or taking time away from "real" research? If so, wait until after you are tenured!
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