Was wondering
Do they exist a general formula for something like this??
√1+√2+√3+√4..........√n
I just thought of it though wanted to ask if there is
looks like you just gave the general formula for the sequence
A_n = √n
For the sum, however, it's not so easy. google "generalized harmonic number"
Thanks sir obleck
Actually the sum was difficult for me
Yes, there is a general formula to calculate the sum of terms in a series like the one you have mentioned: √1 + √2 + √3 + √4 + ... + √n.
To find the general formula, we will use a technique called "telescoping." The idea is to manipulate the series in such a way that most of the terms cancel out, leaving us with a simplified expression.
Let's start by looking at a specific example to understand the pattern:
If we take the first few terms of the series, we have:
√1 + √2 + √3 + √4 + √5 + √6 + √7 + √8 + ...
Now, observe that the difference between consecutive terms can be written as:
(√2 - √1) + (√3 - √2) + (√4 - √3) + (√5 - √4) + (√6 - √5) + (√7 - √6) + (√8 - √7) + ...
Notice that most of the terms cancel each other out:
-√1 from the first term cancels with √1 from the second term.
-√2 from the second term cancels with √2 from the third term.
-√3 from the third term cancels with √3 from the fourth term.
And so on...
After this cancellation, we are left with only the first term (√1) and the last term (√n), which gives us:
Simplified expression = √1 + √n
So, the sum of the terms in the series √1 + √2 + √3 + √4 + ... + √n is equal to √1 + √n.
In conclusion, the general formula for the sum of terms in the given series is √1 + √n.