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1. Which of the following series are convergent?
(i) P∞
n=1
(−1)n−1

n

(ii) P∞
n=1 n
−(
1
2 + 1
n
)

(iii) P∞
n=2
1
n2−1

(iv) P∞
n=1
n
n2+1
You should give a reason in each case.
2. Suppose that f : [0, π/2] → [0, 1] is a continuous function. Show that f(α) = sin α for
some α ∈ [0, π/2].
In the case when f(x) = cos2 x for all x ∈ [0, π/2], determine α to four decimal places.
2. (Original post by sunny0124)
Which of the following series are convergent?
(i) P∞
n=1
(−1)n−1

n

(ii) P∞
n=1 n
−(
1
2 + 1
n
)

(iii) P∞
n=2
1
n2−1

(iv) P∞
n=1
n
n2+1
You should give a reason in each case.
2. Suppose that f : [0, π/2] → [0, 1] is a continuous function. Show that f(α) = sin α for
some α ∈ [0, π/2].
In the case when f(x) = cos2 x for all x ∈ [0, π/2], determine α to four decimal places.
my advice is to post a photo of the question
(saves the typing too)
3. i attached a pdf file for the same.
Attached Images
4. week7.pdf (76.2 KB, 89 views)
5. (Original post by sunny0124)
i attached a pdf file for the same.
(Original post by sunny0124)
Which of the following series are convergent?
(i) P∞
n=1
(−1)n−1

n

(ii) P∞
n=1 n
−(
1
2 + 1
n
)

(iii) P∞
n=2
1
n2−1

(iv) P∞
n=1
n
n2+1
You should give a reason in each case.
2. Suppose that f : [0, π/2] → [0, 1] is a continuous function. Show that f(α) = sin α for
some α ∈ [0, π/2].
In the case when f(x) = cos2 x for all x ∈ [0, π/2], determine α to four decimal places.

These aren't the same at all?
6. The attached isn't the same as the questions so it isn't clear what you want help with. Also this isn't a homework service so you need to post what you have done or what you don't understand etc.
7. (Original post by sunny0124)
i attached a pdf file for the same.
The pdf appears to be some sort of assigment sheet for a university course, which people are NOT going to do for you!

For your original convergence questions you should have covered some convergence tests in lectures, so which ones do you know, what have you applied, and where are you stuck?

As poorform said, this is not a homework service. Post some working / attempts for the questions and someone will try to help you if possible

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