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exe - HiTech-DBF Viewer.Q:

If $\sum_{n=1}^{\infty}\frac{1}{n}=\infty$ then $\lim_{n\to\infty}\frac{1}{n+1}=0$

I'm looking for a way to prove the following statement:

If $\sum_{n=1}^{\infty}\frac{1}{n}=\infty$ then $\lim_{n\to\infty}\frac{1}{n+1}=0$

I've tried the following:

$\frac{1}{n+1}=\frac{1}{n}\cdot\frac{n}{n+1}\leq\frac{1}{n}\cdot\frac{n}{n}=\frac{1}{n}$

$\sum_{n=1}^{\infty}\frac{1}{n}=\infty\Rightarrow\sum_{n=1}^{\infty}\frac{1}{n}\cdot\frac{1}{n}=\sum_{n=1}^{\infty}\frac{1}{n}\cdot\frac{1}{n}=\sum_{n=1}^{\infty}\frac{1}{n}\cdot\frac{1}{n}=\sum_{n=1}^{\infty}\frac{1}{n}\leq\sum_{n=1}^{\infty}\frac{1}{n}=\infty$

Is there any way to prove the statement using the $\epsilon-\delta$-definition of the limit?

A:

By the mean-value theorem for integrals, for any $\epsilon>0$ there is $N$ so that for $n\geq N$, $n+1>n/2$ and so

$ \int_{1}^{n+1}\frac{1}{x}dx\leq\int_{1}^{n/2}\frac{2}{x}dx+\int_{n/2}^{n+1}\frac{2}{x}dx=\frac{2}{n/2}+\frac{2}{n/2}=2. $

Since the partial sums of the harmonic series are convergent, for any $\epsilon>0$ there is $N$ so that for $n\geq N$

$ \frac{1}{n}0$, there is $N$ so that for $n\geq N$

$ \frac{1}{n}Family of mobile-phone-related deaths sue Apple

By IBT Staff Reporter On 04/25/11 AT 6:17 AM

Apple Inc and three South Korean iPhone makers are accused of intentionally slowing down mobile phones to increase their battery life.

Apple did not immediately comment on the lawsuit in the U.S. court. South Korean tech giant Samsung Electronics Co, Chunghwa Telecom Co Ltd and LG Electronics Inc are all named as defendants in the class action suit.

The plaintiffs claim that Apple and the three Korean companies violated South Korea's consumer rights by using a technique called 'variable operating frequency' or VOF.

The plaintiff, Young-jin Kim, also claims that Apple did not adequately inform consumers about the feature on its website and claimed that the VOF process would affect speed, even though the company previously claimed it was designed to enhance battery life.

In his complaint, Kim said that while using an iPhone with VOF on, his phone slowed down from 1.6Ghz to as low as 400MHz within 24 hours.

Kim said that while the VOF technique reduced the power consumption on his phone, he suffered from slower speeds. He said he suffered an upset stomach as a result.

The VOF technology is used by Korean smartphone makers to reduce power consumption. It was introduced by South Korean tech giant Samsung Electronics in the first half of 2010.

The class action suit was filed in U.S. District Court in Los Angeles on Thursday.

Samsung, Chunghwa Telecom and LG Electronics did not immediately comment on the suit.#!/usr/bin/env python

# -*- coding: utf-8 -*-

import json

from alipay.aop.api.constant.ParamConstants import *

class AlipaySocialFeedConfig(object):

def __init__(self):

self._coll_feed = None

@property

def coll_feed(self):

return self._coll_feed

@coll_feed.setter

def coll_feed(self, value):

self._coll_feed = value

def to_alipay_dict(self):

params = dict()

if self.coll_feed:

if hasattr(self.coll_feed, 'to_alipay_dict'):

params['coll_feed'] = self 0b46394aab

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