∑ (from n=1 to ∞) (1 / √(n + 1)) diverges
b. What should...

Question

∑ (from n=1 to ∞) (1 / √(n + 1)) diverges

b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

Accepted Answer

Hello there, today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. For the series, the sum evaluate from i is equal to 1 to infinity of 1 divided by the square root of i plus 1. What is the smallest integer n such that the partial sums subscript n is equal to the sum evaluated from i is equal to 1 ton of 1 divided by the square root of i plus 1. Satisfies S subscript N is greater than 500. OK, so it appears for this particular prompt, we're asked to solve for what will be the smallest integer n value such that the partial sum will be satisfied by S subscript N will be greater than 500. So essentially, we need to take all the information that is provided to us and determine what the value is for the smallest integer value n. So we're solving for lowercase n. Based on the information that's provided to us. So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to state the integral test inequality. So we need to take the integral, evaluate from 1 ton plus 1 of 1 divided by the square root of x + 1. DX is less than or equal to s subscript n, which for simplicity's sake I'll just call SN for short, and this is because our function f of x is equal to 1 divided by the square root of x + 1 is positive. So I'll put a plus sign in quotes, is positive and it is decreasing, so I'll have an arrow pointing down. It's decreasing on. Square bracket one, comma infinity parentheses. So it's decreasing on this specific interval of square bracket one comma, infinity parenthesis. So now, we need to evaluate our integral. So we need to take the integral, evaluate from 1 to N + 1 of 1 divided by the square root of X + 1. DX. And this will be equal to the integral evaluated from 1 ton + 1. Of parentheses X + 1. To the power of 1/2. DX fantastic, we are on a roll here. And this will be. All equal to parentheses x + 1 to the power of 1/2 divided by 1/22, evaluated from 1 ton + 1, which will be equal to 2 multiplied by the square root of x + 1, evaluated from 1 ton + 1. Which will be. Equal to, so this will be equal to 2 multiplied by parentheses of the square root of n + 1 + 1 minus the square root of 1 + 1, which will be equal to 2 multiplied by parentheses of n + 2 minus the square root of 2. So now, at this stage, we need to impose the inequality or exceeding 500, so we need to take 2 multiplied by parentheses of the square root of n + 2 minus the square root of 2 is greater than 500. Which we could write as the square root of n + 2 minus the square root of 2 is greater than 250. The square root of N + 2 is greater than 250 plus the square root of 2. So what we're doing right now is we're manipulating our expression to isolate and solve for N. So I've done a few steps for you to show the methodology, but ultimately, you'll find when you isolate and solve for N all by itself, which I'm gonna skip a few steps because this is basic algebra at this point, we will find that. And Is greater than. 62,500 + 500 multiplied by the square root of 2. Which will be approximately equal to when we round to the nearest whole number. 63,208 And this will be our correct and final answer. Hooray, we did it. And this is when we take the smallest integer that is greater than the bound, and that will be equal to, so that will mean that n. Once again, will be equal to 63,208, and that's it, we've done it. So looking at the prom itself, we can safely say that our final answer for the value of N will be, once again, 63,208, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

∑ (from n=1 to ∞) (1 / √(n + 1)) divergesb. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?

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∑ (from n=1 to ∞) (1 / √(n + 1)) diverges
b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?