- #1
Jenab2
- 85
- 22
Don't ever divide anything by the quantity in the title.
Post your favorite "fancy zeros" here.
Post your favorite "fancy zeros" here.
axmls said:$$\frac{1}{12} + \sum _{n = 1} ^\infty n$$
Jenab2 said:That should probably be
−1/12 + Σ(2,∞) 1/n⁴
Edit: whoops, no. That doesn't seem quite right, either. I evaluated ten million terms of the sum and came up with −0.0010100996222299347, so
−1/12 + 1/999 + Σ(2,∞) 1/n⁴
seems to be nearer to zero.
mfb said:Unless you specify how divergent sums are to be evaluated, the formula is not well-defined.
Yes there is a specific way that leads to -1/12, but this is by far not the only way to assign finite values to divergent sums.
Jenab2 said:1 / Σ(1,∞) n = 0
1 / { 1/a + 1 / Σ(1,∞) n } = a, a≠0.
Ah. My difficulty in appreciating the assignment was caused by my thinking of scalars in vector terms. Consider velocities in the same direction, classically being added, tail to head:axmls said:See here: https://en.m.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯
There are ways to assign a value to that sum using alternate methods. That was the "tongue-in-cheek" aspect of my response.
The value of the given expression is undefined, since the natural logarithm of imaginary numbers (such as i) is undefined. Therefore, the entire expression is undefined.
The possible values of cot [4 arctan 0.2 + (i/2) ln i] range from positive to negative infinity, since the values of arctan 0.2 and ln i can vary between 0 and infinity, resulting in a range of possible values for cot.
No, the expression cannot be simplified as the values of arctan 0.2 and ln i cannot be simplified further. However, if the imaginary component (i) is removed, the expression can be simplified to 9 - cot[4 arctan 0.2] - 1.
The value of the expression will also increase, as the value of cot [4 arctan 0.2 + (i/2) ln i] will increase, resulting in a larger value overall.
Yes, this expression can be evaluated using a calculator, but it is important to use a scientific calculator that can handle complex numbers and has a cot function. Standard calculators may not be able to evaluate this expression accurately.