new .pdf

File information

Original filename: new.pdf

This PDF 1.5 document has been generated by TeX / pdfTeX-1.40.16, and has been sent on pdf-archive.com on 14/03/2017 at 19:04, from IP address 73.204.x.x. The current document download page has been viewed 525 times.
File size: 77 KB (1 page).
Privacy: public file

Download original PDF file

new.pdf (PDF, 77 KB)

Document preview

Another proof that the harmonic series diverges
If the harmonic series converges, then its kth ‘tail’

X
1
T (k) :=
n
n=k+1

is well-defined, and is a strictly decreasing function of k. Now note that
T (2m) =

1
1
1
1
+
+
+
+ ···
2m + 1 2m + 2 2m + 3 2m + 4

1
1
1
1
+
+
+
+ ···
2m + 2 2m + 2 2m + 4 2m + 4

=

2
2
+
+ ···
2m + 2 2m + 4

=

1
1
+
+ ···
m+1 m+2

= T (m),
and if m ≥ 1 then this contradicts that T is a strictly decreasing function.

1

Link to this page

Permanent link

Use the permanent link to the download page to share your document on Facebook, Twitter, LinkedIn, or directly with a contact by e-Mail, Messenger, Whatsapp, Line..

Short link

Use the short link to share your document on Twitter or by text message (SMS)

HTML Code

Copy the following HTML code to share your document on a Website or Blog