# Tagged Questions

2answers
318 views

0answers
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### Rational approximation or series expansion of $K_0$ and $K_1$ for small z

I'm looking for a series expansion of the modified Bessel functions of second kind $K_0(z)$ and $K_1(z)$ for $$|z|<5, ~~|phase(z)| < \pi$$ My $z$ can be described as $z = a\cdot \sqrt{ix}$, ...
3answers
49 views

1answer
178 views

### Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
2answers
3k views

### Power series for $\ln(1+x^2)$

In the problem I am asked to use a power series representation of $\ln(1+x)$ to approximate the integral from $0$ to $0.5$ of $\ln(1+x^2)$ to within 4 decimal places. So far I have found a series for ...
0answers
392 views

0answers
84 views

### Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ Is it related to a known transcendental function? From the research I did, it ...
1answer
201 views

### Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
1answer
348 views

### Sum of power series

Good morning, I have difficulties to find an approximation formula (or bound from the height) for the sum of the following power series $\sum \limits_{k=1}^\infty e^{-k^2}x^k$. Thanks
2answers
104 views

### Approximating $x^k e^{-x}$

I want to approximate the function $f(x) = x^k e^{-x}$ with some finite series. One approach would be to use the power series expansion for $e^{-x}$. But in that case, the power series would have ...
1answer
793 views

### Power series representation of $e^x$ and $e^{-x}$

The power series representation of $e^x = \sum \limits_{k=0}^{\infty} \frac{x^k}{k!}$. Can I use this approximation for $e^{-x} = 1/e^x = 1/\sum \limits_{k=0}^{\infty} \frac{x^k}{k!}$ instead of ...
2answers
503 views

### How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...