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# Sums/Products With Complex Bounds

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I'm bored again.

Almost always, when dealing with sums, the bounds listed are integers. Integer bounds are simple, integer bounds are easy. They can just be added together easily without any complications (example #1). Rational bounds are a bit harder, but they can still be found out by rearranging terms until the fractional bounds become integers (example #2). All other real numbers can be found to a certain precision by truncating the decimal and using the method described previously (example #3). However, one kind of number is left out: complex numbers. Complex numbers are not on the real number line at all. Instead, they are a different kind of number. This makes using methods for real numbers pretty useless when it comes to imaginary numbers. I was stumped for a whole $$30$$ seconds, after which I remembered about a little thing called partial sums. Normally partial sums are used if someone want to find the limit as a boung approaches $$\infty$$ or for finding the exact values of irrational numbers. However, this is one of the very few ways that also works for complex numbers. In other words, I can use partial sums to find values of complex bounds (example #4).

Example #1: $$\sum_{n=0}^{5}(n)=0+1+2+3+4+5=3+3+4+5=6+4+5=10+5=15$$

Example #2: $$\sum_{0}^{2.5}(n)=\frac{1}{4}(\sum_{0}^{5}(n)+2.5)=\frac{0+1+2+3+4+5+2.5}{4}=\frac{15+2.5}{4}=\frac{17.5}{4}=4.375$$

Example #3: $$\sqrt{5}≈2.236~~~~~sorry~the~work~is~too~complicated~to~show~in~a~short~space~~~~~≈3.618$$

Example #4: $$\sum_{n=0}^{5i}(n)=\frac{{(5i)}^{2}+5i}{2}=\frac{5i-25}{2}$$

However, there's a problem with using partial sums to calculate the answer: some sums don't have representable sums. If you work around this problem please comment the answer below. Thanks :)

Feb 22, 2019
edited by Guest  Feb 22, 2019