We wish to find the sum of all positive integers, that is 1+2+3+4+⋯1+2+3+4+⋯. Note we are not looking for a finite sum, but the infinite sum of all integers.
We will try to find a solution using two different methods.
a. First determine ∞∑n=1 n by calculating ∞∑n=1 n − ∞∑n=1(−1)^(n+1) n. According to your results, if we add up all of the infinite integers, do they converge? Is this what you would expect? Why or why not?
b. Now we will take a different approach. Excluding 1, add the sum in groups of threes until you have enough to see a pattern.
1+(2+3+4)+(5+6+7)+⋯1+(2+3+4)+(5+6+7)+⋯.
Use this pattern to find ∞∑n=1 n . According to your result does the series converge or diverge? Compare to your solution for part a.
c. Based on your results, does the sum converge or diverge? Is this what you would expect? Why or why not? Identify any issues or inconsistencies with these calculations. Can we assume that because we can get a mathematical solution that it is valid?
We will try to find a solution using two different methods.
a. First determine ∞∑n=1 n by calculating ∞∑n=1 n − ∞∑n=1(−1)^(n+1) n. According to your results, if we add up all of the infinite integers, do they converge? Is this what you would expect? Why or why not?
b. Now we will take a different approach. Excluding 1, add the sum in groups of threes until you have enough to see a pattern.
1+(2+3+4)+(5+6+7)+⋯1+(2+3+4)+(5+6+7)+⋯.
Use this pattern to find ∞∑n=1 n . According to your result does the series converge or diverge? Compare to your solution for part a.
c. Based on your results, does the sum converge or diverge? Is this what you would expect? Why or why not? Identify any issues or inconsistencies with these calculations. Can we assume that because we can get a mathematical solution that it is valid?
