Is this list a function or a geometric/arithmetic sequence? How do you define it?

Question

When x = 1, y = 1/1
When x = 2, y = 1/1 + 1/2
When x = 3, y = 1/1 + 1/2 + 2/3
When x = 4, y = 1/1 + 1/2 + 2/3 + 3/4
When x = 5, y = 1/1 + 1/2 + 2/3 + 3/4 + 4/5
When x = 6, y = 1/1 + 1/2 + 2/3 + 3/4 + 4/5 + 5/6
When x = 7, y = 1/1 + 1/2 + 2/3 + 3/4 + 4/5 + 5/6 + 6/7
When x = 8, y = 1/1 + 1/2 + 2/3 + 3/4 + 4/5 + 5/6 + 6/7 + 7/8
When x = 9, y = 1/1 + 1/2 + 2/3 + 3/4 + 4/5 + 5/6 + 6/7 + 7/8 + 8/9
When x = 10, y = 1/1 + 1/2 + 2/3 + 3/4 + 4/5 + 5/6 + 6/7 + 7/8 + 8/9 + 9/10

Would this list be a function or a sequence, and how would one go about defining it? I plotted it on Desmos and got this:

Table

Graph

This is $1+\sum\limits_{n=0}^x\frac n{n+1}$ and relates to the harmonic numbers/digamma function — Тyma Gaidash, Jul 24 '23 at 18:55
Or both? "a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position". — peterwhy, Jul 24 '23 at 18:56

score 0 · Accepted Answer · answered Jul 24 '23 at 18:57

You can see the pattern : You obtain the second value of $y$ by adding $\frac{1}{2}$. The one after that is obtained by adding $\frac{2}{3}$ to the previous value. And so on each time adding something of the form $\frac{n}{n+1}$.

You can thus define a sequence $x_n$ for $n\geq 1$ by $x_1=1$ and $x_{n+1}:=x_n+\frac{n}{n+1}$. You can check by hand this agrees with the list you provided. You can also prove this sequence can be expressed as $$ x_n= 1+\sum_{k=1}^n \frac{k-1}{k}.$$

Is this list a function or a geometric/arithmetic sequence? How do you define it?

1 Answers1